Unformatted text preview: TWISTED STABLE MAPS TO TAME ARTIN STACKS arXiv:0801.3040v1 [math.AG] 19 Jan 2008 DAN ABRAMOVICH, MARTIN OLSSON, AND ANGELO VISTOLI Contents 1. Introduction 2. Twisted curves 3. Interlude: Relative moduli spaces 4. Twisted stable maps 5. Reduction of Spaces of Galois admissible covers 6. Example: reduction of X(2) in characteristic 2 Appendix A. Twisted curves and log curves by Martin Olsson Appendix B. Remarks on Extgroups and base change By Martin Olsson Appendix C. Another boundedness theorem for Homstacks By Martin Olsson References 1 3 10 14 20 30 41 49 53 65 1. Introduction This paper is a continuation of [4], where the basic theory of tame Artin stacks is developed. Our main goal here is the construction of an appropriate analogue of Kontsevich’s space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne–Mumford stack, the theory was developed in [1], and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation. However the method of construction we use here is very diﬀerent from the adhoc method of [1], and more natural: we rely on the second author’s general results in [29, 30, 35], as extended in the appendices. Section 2 is devoted to a number of basic properties of twisted curves in arbitrary characteristic, where, unlike the situation of [1], they may fail to
Date : February 2, 2008. Vistoli supported in part by the PRIN Project “Geometria sulle variet` algebriche”, a ﬁnanced by MIUR. Olsson partially supported by NSF grant DMS0555827 and an Alfred P. Sloan fellowship. Abramovich support in part by NSF grants DMS0301695 and DMS0603284.
1 2 ABRAMOVICH, OLSSON, AND VISTOLI be Deligne–Mumford stacks. In particular we show in Proposition 2.3 that the notion of twisted curve is stable under small deformations and can be tested on geometric ﬁbers. In Section 3 we collect together some facts about relative moduli spaces which will be used in the following section. In Section 4 we deﬁne twisted stable maps into a tame stack X and show in Theorem 4.1 that they form an artin stack, which is proper and quasiﬁnite over the Kontsevich space of the coarse moduli space X of X . In particular, when X is projective, the stacks of twisted stable maps of X admit projective coarse moduli spaces. The construction proceeds rather naturally from the following: (1) the existence of the stack of twisted curves, which was shown in [29] for Deligne–Mumford twisted curves and proved in our case in Appendix A, and (2) The existence and ﬁniteness properties of Homstacks, proved in [30] and [35] in many cases and extended in our case in Appendix C. Some extra care is needed for proving the quasiﬁnite claim. Properness relies on Lemma 4.4, a suitable generalization of the Purity Lemma of [1]. In section 5 we concentrate on the case where the target stack is B G, the classifying stack of a ﬁnite ﬂat linearly reductive group scheme G. The main result here, Theorem 5.1, is that the space of stable maps with target B G is ﬁnite and ﬂat over the corresponding Deligne–Mumford space. This result is known when G is tame and ´tale (see [2]) and relatively straightforward e when G is a diagonalizable group scheme (or even a twist of such). However, our argument in general takes some delicate twists and turns. As an example for the behavior of these stacks, we consider in Section 6 two ways to compactify the moduli space X (2) of elliptic curves with full level2 structure. The ﬁrst is as a component in K0,4 (B µ2 ), the stack of totally branched µ2 covers of stable 4pointed curves of genus 0. This provides an opportunity to consider the cyclotomic inertia stacks and evaluation maps. The second is as a component in K1,1 (B µ2 ) parametrizing 2 elliptic curves with B µ2 , where X (2) meets other components in a geomet2 rically appealing way. We also ﬁnd the KatzMazur regular model of X (2) as the closure of a component of the generic ﬁber. As already mentioned, the paper contains three appendices, written by the second author. Appendix A generalizes the main results of [29]. Appendix B contains some prepatory results needed for Appendix C which generalizes some of the results from [30] and [35] to tame stacks. The logical order of Appendix A is after section 2, whereas the Appendix C only uses results from [4]. 1.1. Acknowledgements. Thanks to Johan de Jong for helpful comments. Thanks to Shaul Abramovich for advice with visualization. TWISTED YET TAME 3 2. Twisted curves Deﬁnition 2.1. Let S be a scheme. An nmarked twisted curve over S is a collection of data (f : C → S, {Σi ⊂ C}n ) as follows: i=1 (i) C is a proper tame stack over S whose geometric ﬁbers are connected of dimension 1, and such that the moduli space C of C is a nodal curve over S . (ii) The Σi ⊂ C are closed substacks which are fppf gerbes over S , and whose images in C are contained in the smooth locus of the morphism C → S. (iii) If U ⊂ C denotes the complement of the Σi and the singular locus of C → S , then U → C is an open immersion. (iv) For any geometric point s → C mapping to a smooth point of C , there ¯ exists an integer r such that Spec(OC,s ) ×C C ≃ [Dsh /µr ], ¯ where D sh denotes the strict henselization of D := Spec(OS,f (¯) [z ]) at s the point (mS,f (¯) , z ) and ζ ∈ µr acts by z → ζ · z (note that r = 1 s unless s maps to a point in the image of some Σi ). Here mS,f (¯) denote ¯ s the maximal ideal in the strict henselization OS,f (¯) . s (v) If s → C is a geometric point mapping to a node of C , then there exists ¯ an integer r and an element t ∈ mS,f (¯) such that s Spec(OC,s ) ×C C ≃ [Dsh /µr ], ¯ where Dsh denotes the strict henselization of D := Spec(OS,f (¯) [z, w]/(zw − t)) s at the point (mS,f (¯) , z, w) and ζ ∈ µr acts by x → ζ · z and y → ζ −1 · y . s Remark 2.2. If C → S is a proper tame Artin stack which admits a collection of closed substacks Σi ⊂ C (i = 1, . . . , n for some n) such that (C , {Σi }n ) is an nmarked twisted curve, we will refer to C as a twisted i=1 curve, without reference to markings. The following proposition allows us to detect twisted curves on ﬁbers: Proposition 2.3. Let S be a scheme, and let (f : C → S, {Σi }n ) be a i=1 proper ﬂat tame stack f : C → S with a collection of closed substacks Σi ⊂ C which are S gerbes. If for some geometric point x → S the ﬁber (Cx , {Σi,x }) ¯ ¯ ¯ is an nmarked twisted curve, then there exists an open neighborhood of x ¯ in S over which (C → S, {Σi }) is an nmarked twisted curve. Proof. Let π : C → C be the coarse moduli space of S . Since C is a tame stack ﬂat over S , the space C is ﬂat over S , and for any morphism S ′ → S the base change C ×S S ′ is the coarse moduli space of C ×S S ′ , by [4, Corollary 3.3]. It follows that in some ´tale neighborhood of x → S the space C/S e ¯ is a nodal curve over S (see for example [11, §1]). Shrinking on S we may therefore assume that C is a nodal curve over S . After further shrinking on 4 ABRAMOVICH, OLSSON, AND VISTOLI S , we can also assume that the images of the Σi in C are contained in the smooth locus of the morphism C → S and that (2.1 (iii)) holds. To prove the proposition it then suﬃces to verify that conditions (iv) and (v) hold for geometric points of s → C over x. ¯ ¯ We may without loss of generality assume that S is the spectrum of a strictly henselian local ring whose closed point is x. Also let Cs denote the ¯ ¯ ﬁber product Cs := C ×C,s Spec(OC,s ). ¯ ¯ ¯ Consider ﬁrst the case when s → C has image in the smooth locus of C/S . ¯ Since the closed ﬁber Cx is a twisted curve, we can choose an isomorphism ¯ Cs ×S Spec(k(¯)) ≃ [Spec(k(¯)[z ])sh /µr ], x x ¯ for some integer r ≥ 1, where the µr action is as in (2.1 (iv)). Let P0 → Cs ×S Spec(k(¯)) x ¯ be the µr torsor Spec(k(¯)[z ])sh → [Spec(k(¯)[z ])sh /µr ]. x x As in the proof of [4, Proposition 3.6] there exists a µr torsor P → Cs whose ¯ reduction is P0 . Furthermore, P is an aﬃne scheme over S . Since Cs is ﬂat ¯ over S , the scheme P is also ﬂat over S . Since its closed ﬁber is smooth over S , it follows that P is a smooth S –scheme of relative dimension 1 with µr action. Write P = Spec(A) and ﬁx a µr equivariant isomorphism A ⊗OS k(¯) ≃ (k(¯)[z ])sh . x x Since µr is linearly reductive we can ﬁnd an element z ∈ A lifting z such ˜ that an element ζ ∈ µr acts on z by z → ζ · r . Since A is ﬂat over OS the ˜ ˜ ˜ induced map (OS [z ])sh → A is an isomorphism, which shows that (iv) holds. The veriﬁcation of (v) requires some deformation theory. First we give an explicit argument. We then provide an argument using the cotangent complex. Let s → C be a geometric point mapping to a node of C . Then we can ¯ write Cs = [Spec(A)/µr ], ¯ for some integer r , and A is a ﬂat OS,x algebra such that there exists an ¯ isomorphism ι : A ⊗OS,x k(¯) ≃ (k(¯)[z, w]/(zw))sh , x x ¯ with µr action as in (v). We claim that we can lift this isomorphism to a µr equivariant isomorphism (2.3.1) A ≃ (OS,x [˜, w ]/(˜w − t))sh z˜ ¯z ˜ for some t ∈ OS,x . For this note that Aµr is equal to OC,s and that A is a ¯ ¯ ﬁnite OC,s algebra. By a standard application of the Artin approximation ¯ TWISTED YET TAME 5 theorem, in order to ﬁnd 2.3.1 it suﬃces to prove the analogous statements for the map on completions (with respect to mC,s ) ¯ OC,s → A. ¯ For this in turn we inductively ﬁnd an element tq ∈ OS,x /mq x and a ¯ S, ¯ µr equivariant isomorphism ρq : ((OS,x /mq x )[z, w]/(zw − tq ))∧ → A/mq x A. ¯ S, ¯ S, ¯ For ρ1 we take the isomorphism induced by ι and t1 = 0. For the inductive step we assume (ρq , tq ) has been constructed and ﬁnd (ρq+1 , tq+1 ). For this choose ﬁrst any liftings z , w ∈ A/mq+1 A of z and w such that ζ ∈ µr acts ˜˜ S,x ¯ by (2.3.2) z → ζ · z , w → ζ −1 · w. ˜ ˜˜ ˜ This is possible because µr is linearly reductive. Since A is ﬂat over OS,x we have an exact sequence ¯ 0 → mq x /mq+1 ⊗k(¯) A/mS,x → A/mq+1 A → A/mq x A → 0. ¯ x S, ¯ S,x ¯ S,x ¯ S, ¯ Choosing any lifting tq+1 ∈ OS,x /mq+1 of tq and consider ¯ S,x ¯ z w − tq+1 . ˜˜ This is a µr invariant element of mq x /mq+1 ⊗k(¯) A/mS,x ≃ mq x /mq+1 ⊗k(¯) ((OS,x /mS,x )[z, w]/(zw))∧ . ¯ ¯ ¯ x x S, ¯ S,x ¯ S, ¯ S,x ¯ It follows that after possibly changing our choice of lifting tq+1 of tq , we can write z w = tq+1 + z r g + wr h, ˜˜ ˜ ˜ where tq+1 ∈ OS,x /mq+1 and reduces to tq , and g, h ∈ mq x A/mq+1 A are ¯ S,x ¯ S, ¯ S,x ¯ µr invariant. Replacing z by z − wr−1 h and w by w − z r−1 g (note that with ˜ ˜˜ ˜ ˜˜ these new choices the action of µr is still as in 2.3.2) we obtain tq+1 and ρq+1 . ♠ We give an alternative proof of (v) using a description of the cotangent complex of C , which might be of interest on its own. Since C is an Artin stack a few words are in order. The cotangent complex LX /Y of a morphism of Artin stacks X → Y is deﬁned in [24], Chapter 17 as an object in the derived category of quasicoherent sheaves in the lisse´tale site of X . Unfortunately, e as was observed by Behrend and Gabber, this site is not functorial, which had the potential of rendering LX both incomputable and useless. However, in [28], see especially section 8, it is shown that the lisse´tale site has e surrogate properties replacing functoriality which are in particular suﬃcient ′ for dealing with LX , as an object of a carefully constructed Dqcoh (Xlis´t ), e and with deformation theory. Two key properties are the following: 6 ABRAMOVICH, OLSSON, AND VISTOLI (1) given a morphism f : X → Y of algebraic stacks over a third Z , there is a natural distinguished triangle Lf ∗ LY /Z → LX /Z → LX /Y → . (2) Given a ﬁber diagram X′
f [1] // X // Y Y′ with the horizontal maps ﬂat, we have f ∗ LX /Y = LX ′ /Y ′ . If the base scheme is S , consider S → B Gm,S . Applying the above to the ﬁber square // S Gm,S S // B Gm,S and the morphisms S → B Gm,S → S it is easy to see that LBGm,S /S = OBGm,S [−1]. (In fact for a general smooth group scheme G we have LS/BG = Lie(G)∗ and therefore LBG/S = Lie(G)∗ [−1] considered as a Gequivariant object.) Let now S = Spec k with k separably closed, and consider a twisted curve C /S with coarse moduli space π : C → C . Let s → C be a geometric point ¯ mapping to a node, and ﬁx an isomorphism Cs := C ×C s ≃ [Dsh /µr ], ¯ ¯ where D := Spec(k[z, w]/(zw)). By a standard limit argument, we can thicken this isomorphism to a diagram
~ α ~~ ~ ~ ~ ~~ CU G G
πU GG β GG GG C ## [D/µr ] π a C } }} }} ~~}} U GG } GG b GG GG G ## D µr , where Dµr denote the coarse space of [D/µr ] (so D µr is equal to the spectrum of the invariants in k[z, w]/(zw)), a and b are ´tale, U is aﬃne, and the e squares are cartesian. In this setting we can calculate a versal deformation of CU as follows. TWISTED YET TAME 7 First of all we have the deformation [Spec(k[z, w, t]/(zw − t))/µr ] of [D/µr ]. Since β is ´tale and representable and by the invariance of the e ´tale site under inﬁnitesimal thickenings, this also deﬁnes a formal deformae tion (i.e. compatible family of deformations over the reductions) CU,t → Spf(k[[t]]) of CU . We claim that this deformation is versal. Since this deformation is nontrivial modulo t2 , to verify that CU,t is versal it suﬃces to show that the deformation functor of CU is unobstructed and that the tangent space is 1dimensional. For this it suﬃces to show that Ext2 (LCU , OC ) = 0 and Ext1 (LCU , OC ) = k. The map CU → [D/µr ] induces a morphism CU → B µr . Consider the composite map CU → B µr → B Gm . Since LBGm /k ≃ OS [−1] we then obtain a distinguished triangle OC [−1] // LCU // LCU /BGm [1] // . Considering the associated long exact sequence of Extgroups one sees that Exti (LCU , OC ) = Exti (LCU /BGm , OC ) for i > 0. Now to prove that Ext2 (LCU /BGm , OC ) = 0, and Ext1 (LCU /BGm , OC ) = k, it suﬃces to show that LC /BGm can be represented by a twoterm complex F −1 → F 0 with the F i locally free sheaves of ﬁnite rank and the cokernel of the map Hom(F 0 , OCU ) → Hom(F −1 , OCU ) isomorphic to the structure sheaf OZ of the singular locus Z ⊂ CU (with the reduced structure). Now since β is ´tale, the complex LCU /BGm is isomorphic to β ∗ L[D/µr ]/BGm . e Therefore to construct such a presentation F · of LCU /BGm , it suﬃces to construct a corresponding presentation of L[D/µr ]/BGm . For this consider the ﬁber diagram // C X S // B Gm Here X is the surface Spec(k[z ′ , w′ , u, u−1 ]/(z ′ w′ )) (the action of Gm is via t · (z ′ , w′ , u) = (tz ′ , t−1 w′ , tr u)). Since X/S is a local complete intersection, the 8 ABRAMOVICH, OLSSON, AND VISTOLI map C → B Gm is also a representable local complete intersection morphism, and therefore the natural map LC /BGm → H0 (LC /BGm ) = Ω1 /BGm is a quasiC isomorphism. Concretely, the pullback of LC /BGm is isomorphic to Ω1 . X/S Now we have a standard twoterm resolution
3 0 → OX → OX → Ω1 → 0 X/S where the ﬁrst map is given by (w′ , z ′ , 0) and the second by (dz ′ , dw′ , du/u). Our desired presentation of L[D/µr ]/BGm is then obtained by noting that this resolution clearly descends to a locally free resolution on C . Remark 2.4. Let S be a scheme. A priori the collection of twisted npointed curves over S (with morphisms only isomorphisms) forms a 2category. However, by the same argument proving [1, 4.2.2] the 2category of twisted npointed curves over S is equivalent to a 1category. In what follows we will therefore consider the category of twisted npointed curves. Proposition 2.5. Let f : C → S be a twisted curve. Then the adjunction map OS → f∗ OC is an isomorphism, and for any quasicoherent sheaf F on C we have Ri f∗ F = 0 for i ≥ 2. ¯ Proof. Let π : C → C be the coarse space, and let f : C → S be the structure morphism. Since π∗ is an exact functor on the category of quasicoherent sheaves on C , we have for any quasicoherent sheaf F on C an equality ¯ Ri f∗ F = Ri f∗ (π∗ F ). ¯ Since f : C → S is a nodal curve this implies that Ri f∗ F = 0 for i ≥ 2, and the ﬁrst statement follows from the fact that π∗ OC = OC . ♠ We conclude this section by recording some facts about the Picard functor of a twisted curve (these results will be used in section 5 below). Let S be a scheme and f : C → S a twisted curve over S . Let PicC /S denote the stack over S which to any S scheme T associates the groupoid of line bundles on the base change CT := C ×S T . Proposition 2.6. The stack PicC /S is a smooth algebraic stack over S , and for any object L ∈ PicC /S (T ) (a line bundle on CT ), the group scheme of automorphisms of L is canonically isomorphic to Gm,T . Proof. That PicC /S is algebraic is a standard application of Artin’s criterion for verifying that a stack is algebraic [5], and can also be seen as a Homstack. See [6] for details. To see that PicC /S is smooth we apply the inﬁnitesimal criterion. For this it suﬃces to consider the case where S aﬃne and we have a closed subscheme S0 ⊂ S with square0 ideal I . Writing C0 for C ×S S0 , we have an exact sequence 0 → OC ⊗ I → OC → OC0 → 0 giving an exact sequence of cohomology PicC /S (S ) → PicC /S (S0 ) → R2 f∗ OC ⊗ I. TWISTED YET TAME 9 By Proposition 2.5 the term on the right vanishes, giving the existence of the desired lifting of PicC /S (S ) → PicC /S (S0 ). The statement about the group of automorphisms follows from the fact that the map OS → f∗ OC is an isomorphism, and the same remains true after arbitrary base change T → S . ♠ Let PicC /S denote the rigidiﬁcation of PicC /S with respect to the group scheme Gm [4, Appendix A]. Let π : C → C denote the coarse moduli space of C (so C is a nodal curve over S ). Pullback deﬁnes a fully faithful functor π ∗ : PicC/S → PicC /S which induces a morphism (2.6.1) π ∗ : PicC/S → PicC /S . This morphism is a monomorphism of group schemes over S . Indeed suppose given two S valued points [L1 ], [L2 ] ∈ PicC/S (S ) deﬁned by line bundles L1 and L2 on C such that π ∗ [L1 ] = π ∗ [L2 ]. Then after possibly replacing S by an ´tale cover, the two line bundles π ∗ L1 and π ∗ L2 on C are isomorphic. e Since Li = π∗ π ∗ Li (i = 1, 2) this implies that L1 and L2 are also isomorphic. Lemma 2.7. The cokernel W of the morphism π ∗ in (2.6.1) is an ´tale e group scheme over S . Remark 2.8. Note that, in general, W is nonseparated. Consider S the aﬃne line over a ﬁeld k with parameter t, and consider the blowup X of P1 × S at the origin. The zero ﬁber has a singularity with local equation uv = t. There is a natural action of µr along the ﬁber, which near the − singularity looks like (u, v ) → (ζr u, ζr 1 v ). The quotient C = [X/µr ] is a twisted curve, with twisted markings at the sections at 0 and ∞, and a twisted node. The coarse curve C = [X/µr ] has an Ar−1 singularity xy = tr , where x = ur and y = v r . Let E ⊂ C be one component of the singular ﬁber. The invertible sheaf OC (E ) gives a generator of W , but it coincides with the trivial sheaf away from the singular ﬁber. In fact W is the groupscheme obtained by gluing r copies of S along the open subset t = 0. Proof. The space W is the quotient of a smooth group scheme over S by the action of a smooth group scheme, hence is smooth over S . The fact that W is ´tale amounts (using the inﬁnitesimal lifting property) to the statement e that if T0 ֒→ T is a square zero nilpotent thickening deﬁned by a sheaf of ideals I ⊂ OT , and if L is a line bundle on CT whose reduction L0 on CT0 is obtained by pullback from a line bundle on CT0 , then L is the pullback of a line bundle on CT . Using the exponential sequence as usual this amounts to the statement that the map H1 (CT0 , I OCT ) → H1 (CT0 , I OCT ) is an isomorphism. ♠ 10 ABRAMOVICH, OLSSON, AND VISTOLI Lemma 2.9. The cokernel W is quasiﬁnite over S . Proof. We may assume that S is aﬃne, say S = Spec A. Then the twisted cuver C is deﬁned over some ﬁnitely generated Zsubalgebra A0 , and formation of W commutes with base change; hence we may assume that A = A0 . In particular S is noetherian. We may also assume that S is reduced. After passing to a stratiﬁcation of S and to ´tale coverings, we may assume that S is connected, that C → S e is of constant topological type, that the nodes of the geometric ﬁbers of C → S are supported along closed subschemes S1 , . . . , Sk of C which map isomorphically onto S , and the reduced inverse image Σi of Si in C is isomorphic to the classfying stack B µmi for certain positive integers m1 , . . . , mk . For any line bundle L on C the pullback of L to Σi is a line bundle Li on Si , with an action of µmi , given by a character µmi → Gm of the form t → tai for some ai ∈ Z/(mi ). By sending L into the collection (ai ) we obtain a morphism PicC /S → i Z/(mi ), whose kernel is easily seen to be PicC/S . Therefore we have a categorically injective map W → i Z/(mi ); since W and i Z/(mi ) are ´tale over S , this is an open embedding. Since e Z/(mi ) is noetherian and quasiﬁnite over S , so is W . ♠ i If N denotes an integer annihilating W we obtain a map ×N : PicC /S → PicC/S . Deﬁnition 2.10. Let Pico /S denote the ﬁber product C Pico /S = PicC /S ××N,PicC/S Pico . C C/S The open and closed subfunctor Pico /S ⊂ PicC /S classiﬁes degree 0line C bundles on C . In particular, Pico /S is independent of the choice of N in the C above construction. Note also that any torsion point of PicC /S is automatically contained in Pico /S since the cokernel of C Pico → PicC/S C/S is torsion free. Since Pico is a semiabelian scheme over S , we obtain the following: C/S Corollary 2.11. The group scheme Pico /S is an extension of a quasiﬁnite C ´tale group scheme over S by a semiabelian group scheme. e 3. Interlude: Relative moduli spaces In this section we gather some results on relative moduli spaces which will be used in the veriﬁcation of the valuative criterion for moduli stacks of twisted stable maps in the next section. For a morphism of algebraic stacks f : X → Y , denote by I (X /Y ) = Ker (I (X ) → f ∗ I (Y )) = X ×X ×Y X X the relative inertia stack. TWISTED YET TAME 11 Now let f : X → Y be a morphism of algebraic stacks, locally of ﬁnite presentation, and assume the relative inertia I (X /Y ) → X is ﬁnite. Recall that when Y is an algebraic space we know that the morphism factors through the coarse moduli space: X → X → Y , see [21, 10]. We claim that there is a relative construction that generalizes to our situation: Theorem 3.1. There exists an algebraic stack X , and morphisms X → X → Y such that f = f ◦ π , satisfying the following properties: (1) f : X → Y is representable, (2) X → X → Y is initial for diagrams satisfying (1), namely if X → X ′ → Y has representable f ′ then there is a unique h : X → X ′ such ′ that π ′ = h ◦ π and f = f ◦ h, (3) π is proper and quasiﬁnite, (4) OX → π∗ OX is an isomorphism, and (5) if X ′ is a stack and X ′ → X is a representable ﬂat morphism, then X ′ → Y is the relative coarse moduli space of X ×X X ′ → Y . Proof. Consider a smooth presentation R ⇉ U of Y . Denote by XR the coarse moduli space of XR = X ×Y R and XU the coarse moduli space of XU = X ×Y U , both exist by the assumption of ﬁniteness of relative inertia. Since the formation of coarse moduli spaces commutes with ﬂat base change, we have that XR = XU ×U R. It is straightforward to check that XR ⇉ UR is a smooth groupoid. Denote its quotient stack X := [XR ⇉ UR ]. It is again straightforward to construct the morphisms X → X → Y from the diagram of relevant groupoids. Now X ×Y U → XU is an isomorphism, U → Y is surjective and XU → U is representable, hence X → Y is representable, giving (1). Similarly XU → XU is proper and quasiﬁnite hence X → X is proper and quasiﬁnite, giving (3). Also (4) follows by ﬂat basechange to XU . Part (5) follows again since formation of moduli spaces commutes with ﬂat base change: the coarse moduli space of XU ×X X ′ is XU ×X X ′ and similarly the coarse moduli space of XR ×X X ′ is XR ×X X ′ . Now consider the situation in (2). Since X ′ → Y is representable, it is presented by the groupoid X ′ ×Y R ⇉ X ′ ×Y U , which, by the universal property of coarse moduli spaces, is the target of a canonical morphism of groupoids from XR ⇉ UR , giving the desired morphism h : X → X ′ . ♠ We now consider the tame case. We say that a morphism f : X → Y with ﬁnite relative inertia is tame if for some algebraic space U and faithfully ﬂat U → Y we have that X ×Y U is a tame algebraic stack. This notion is independent of the choice of U by [4, Theorem 3.2]. Let X → X → Y be the relative coarse moduli space.
π f f′ π f π′ f π 12 ABRAMOVICH, OLSSON, AND VISTOLI Proposition 3.2. If X ′ is an algebraic stack and X ′ → X is a representable morphism of stacks, then X ′ → Y is the relative coarse moduli space of X ×X X ′ → Y . This is proven as in part (5) of the theorem, using the fact that formation of coarse moduli spaces of tame stacks commutes with arbitrary base change. We now consider a special case which is relevant for our study of tame stacks. Let V be a strictly henselian local scheme, with a ﬁnite linearly reductive group scheme Γ acting, ﬁxing a geometric point s → V . Let X = [V /Γ], and consider a morphism f : X → Y , with Y also tame. Let K ⊂ Γ be the subgroup scheme ﬁxing the composite object s → V → Y (so K is the kernel of the map Γ → AutY (f (s))). Consider the geometric quotient U = V /K , the quotient group scheme Q = Γ/K and let X = [U/Q]. There is a natural projection q : X → X . Proposition 3.3. There exists a unique morphism g : X → Y such that f = g ◦ q , and this factorization of f identiﬁes X as the relative coarse moduli space of X → Y . Proof. Consider the commutative diagram
f f′ h XB B // YX X // Y // Y BB BB BB f where X, Y are the coarse moduli spaces of Y and Y , respectively, the square is cartesian, and f is induced from f . Since h is fully faithful we may replace Y by YX . As the formation of coarse moduli space of tame stacks commutes with arbitrary base change, X is the coarse moduli space of YX . We may therefore assume that X = Y . It also suﬃces to prove the proposition after making a ﬂat base change on X . We may therefore assume that the residue ﬁeld of V is algebraically closed. We can write Y = [P/G], where P is a ﬁnite X scheme and G is a ﬁnite ﬂat split linearly reductive group scheme over X . We can assume P is local strictly henselian as well, and that the action of G on P ﬁxes the closed point p ∈ P . The map f induces, by looking at residual gerbes, a map Bρ0 : BΓ0 → BG0 where Γ0 and G0 denote the corresponding reductions to the closed point of X . This is induced by a group homomorphism ρ0 : Γ0 → G0 , uniquely deﬁned up to conjugation. Since Γ and G are split, the homomorphism ρ0 lifts to a homomorphism ρ : Γ → G. TWISTED YET TAME 13 We obtain a 2commutative diagram BΓ0
s ¯ Bρ0 // BG0
p ¯ [V /Γ] f // [P/G]. Let P ′ → [V /Γ] be the pullback of the Gtorsor P → [P/G] Lemma 3.4. There is an isomorphism of Gtorsors over [V /Γ] [V × G/Γ] → P ′ . Proof. Note that V ×[V /Γ] BΓ0 = s and P ′ ×[V /Γ] BΓ0 = BΓ0 ×BG0 p. Then the commutativity of the diagram s // p // BG0 BΓ0 implies that there is an isomorphism of G0 torsors over BΓ0 ([V × G/Γ] ×[V /Γ] BΓ0 ∼ // P ′ ×[V /Γ] BΓ0 .
♠ As in the proof of [4, 3.6], this lifts to an isomorphism as required. Now we have a morphism
˜ f V // [V × G/Γ] // P ′ &&// P compatible with the actions of Γ and G. By the universal property of the ˜ quotient, the map f factors through U . Passing to the quotient by the respective group actions, we get morphisms [V /Γ] → [U/Q] → [P/G]. Let X → Z → Y be the relative coarse moduli space. Since Q injects in G, the morphism [U/Q] → [P/G] is representable, and therefore we have a morphism Z → [U/Q] over Y . To check this is an isomorphism it suﬃces to check after base change along the ﬂat morphism P → Y . In other words, we need to check that [V /Γ] ×[P/G] P → [U/Q] ×[P/G] P is the coarse moduli space. This map can be identiﬁed with [V × G/Γ] → U ×Q G. This follows by noting that (OV ⊗ OG )Γ = ((OV )K ⊗ OG )Q = (OU ⊗ OG )Q . ♠ 14 ABRAMOVICH, OLSSON, AND VISTOLI 4. Twisted stable maps This section relies on the results of Appendix A, in which the stack of twisted curves is constructed. Let M be a ﬁnitely presented tame algebraic stack with ﬁnite inertia proper over a scheme S . A twisted stable map from an nmarked twisted curve (f : C → S, {Σi ⊂ C}n ) over S to M is a representable morphism of i=1 S stacks g:C→M such that the induced morphism on coarse spaces g:C→M is a stable map (in the usual sense of Kontsevich) from the npointed curve (C, {p1 , . . . , pn }n ) to M (where the points pi are the images of the Σi ). i=1 If M is projective over a ﬁeld, the target class of g is the class of g∗ [C ] in the chow group of 1dimensional cycles up to numerical equivalence of M . Theorem 4.1. Let M be a ﬁnitely presented tame algebraic stack proper over a scheme S with ﬁnite inertia. Then the stack Kg,n (M) of twisted stable maps from npointed genus g curves into M is a locally ﬁnitely presented algebraic stack over S , which is proper and quasiﬁnite over the stack of stable maps into M . If M is projective over a ﬁeld, the substack Kg,n (M, β ) of twisted stable maps from npointed genus g curves into M with target class β is proper, and is open and closed in Kg,n (M). Proof. The statement is local in the Zariski topology of S , therefore we may assume S aﬃne. Since M is of ﬁnite presentation, it is obtained by base change from the spectrum of a noetherian ring, and replacing S by this spectrum we may assume M is of ﬁnite type over a noetherian scheme S . As M is now of ﬁnite type, there is an integer N such that the degree of the automorphism group scheme of any geometric point of M is ≤ N . Consider the stack of twisted curves Mtw deﬁned in A.6. It contains an g,n tw , open substack Mg,n≤N ⊂ Mtw of twisted curves of index bounded by N g,n tw , by A.8. The natural morphism Mg,n≤N → Mg,n to the stack of prestable curves is quasiﬁnite by A.8. Consider B = Mtw,≤N ×Spec Z S . It is an algebraic stack of ﬁnite type g,n over B = Mg,n ×Spec Z S . Denote MB = M ×S B, MB = M ×S B and similarly MB = M ×S B . These are all algebraic stacks. Finally consider the universal curves Ctw → Mtw and C → Mg,n , and g,n their corresponding pullbacks Ctw , CB and CB . B As discussed in Appendix C, there is an algebraic stack Homrep (Ctw , MB), B B locally of ﬁnite type over B, parametrizing representable morphisms of the universal twisted curve to M. We also have the analogous HomB(CB, MB) TWISTED YET TAME 15 and HomB (CB , MB ). We have the natural morphisms (4.1.1) Homrep (Ctw , MB) B B
a // HomB(CB, MB) b // HomB (CB , MB ). Proposition 4.2. The morphism a is quasiﬁnite. Proof. It suﬃces to prove the following. Let k be an algebraically closed ﬁeld with a morphism Spec(k) → S and C /k a twisted curve with coarse moduli space C . Let f : C → M be a S morphism, and let G → C denote the pullback of M to C along the composite C // C f // M. We then need to show that the groupoid Sec(G /C )(k) (notation as in C.17) has only ﬁnitely many isomorphism classes of objects. Let C ′ denote the normalization of C (so C ′ is a disjoint union of smooth curves), and let C ′ → C be the maximal reduced substack of the ﬁber product C ×C C ′ . For any node x ∈ C let Σx denote maximal reduced substack of Cx (so Σx is isomorphic to Bµr for some r ), and let Σ denote the disjoint union of the Σx over the nodes x. Then there are two natural inclusions j1 , j2 : Σ ֒→ C ′ , and by [3, A.0.2] the functor Hom(C , M)(k) ∗ ∗ Hom(C ′ , M)(k) ×j1 ×j2 ,Hom(Σ,M)(k)×Hom(Σ,M)(k),∆ Hom(Σ, M)(k) is an equivalence of categories. Since the stack Hom(Σ, M) clearly has quasiﬁnite diagonal, it follows that the map Sec(G /C ) → Sec(G ×C C ′ /C ′ ) is quasiﬁnite. It follows that to prove 4.2 it suﬃces to consider the case when C is smooth. Let Y denote the normalization of G . Then any section C → G factors uniquely through Y so it suﬃces to show that the set Sec(Y /C )(k) is ﬁnite. To prove this we may without loss of generality assume that there exists a section s : C → Y . For any smooth morphism V → C the coarse space of the ﬁber product GV := G ×C V is equal to V by [4], Corollary 3.3 (a). For a smooth morphism V → C , let d // c // YV V Y
V be the factorization of YV → V provided by [35, 2.1] (rigidiﬁcation of the generic stabilizer). The section s induces a section V → Y V , and hence by [30, 2.9 (ii)] the projection d : Y → V is in fact an isomorphism. It follows 16 ABRAMOVICH, OLSSON, AND VISTOLI that Y is a gerbe over C . If G → C denotes the automorphism group scheme of s, then in fact Y = BC G. Let ∆ ⊂ G denote the connected component of G, and let H denote the ´tale part of G, so we have a short exact sequence e 1→∆→G→H→1 of group schemes over C . Let C ′ → C be a ﬁnite ´tale cover such that H and the Cartier dual of ∆ e constant. If C ′′ denotes the product C ′ ×C C ′ and Y ′ and Y ′′ the pullbacks of Y to C ′ and C ′′ respectively, then one sees using a similar argument to the one used in [30, §3] that it suﬃces to show that Sec(Y ′ /C ′ )(k) and Sec(Y ′′ /C ′′ )(k) are ﬁnite. We may therefore assume that H and the Cartier dual of ∆ are constant. The map G → H induces a projection over C BC G → BC H. Giving a section z : C → BC H is equivalent to giving an Htorsor P → C . Since C is normal and H is ´tale, such a torsor is determined by its restriction e to any dense open subspace of C . Since C contains a dense open subscheme and the ´tale fundamental group of such a subscheme is ﬁnitely generated e it follows that Sec(BC H/C )(k) is ﬁnite. Fix a section z : C → BC H, and let Q → C be the ﬁber product Q := C ×z,BC H BC G. Then Q is a gerbe over C banded by a twisted form of ∆. By replacing C by a ﬁnite ´tale covering as above, we may assume that in fact Q is a e gerbe banded by ∆ and hence isomorphic to C ×Spec(k) B ∆0 for some ﬁnite diagonalizable group scheme ∆0 /k. Writing ∆0 as a product of µr ’s, we even reduce to the case when ∆0 = µr . In this case, as explained in C.25, giving a section w : C → BC ∆ is equivalent to giving a pair (L, ι), where L is a line bundle on C and ι : OC → L⊗r is an isomorphism. We conclude that the set of sections C → Q is in bijection with the set PicC /k [r ](k) of r torsion points in the Picard space PicC /k . By 2.11 this is a ﬁnite group, and hence this completes the proof of 4.2. ♠ Consider again the diagram 4.1.1. The morphism a is quasiﬁnite by 4.2 and of ﬁnite type by C.7. The second morphism is quasiﬁnite and of ﬁnite type as it is obtained by base change. Therefore the composite 4.1.1 is also quasiﬁnite and of ﬁnite type. TWISTED YET TAME 17 The Kontsevich moduli space Mg,n (M ) is open in HomB (CB , MB ). We have Kg,n (M) ≃ Homrep (Ctw , MB) ×HomB (CB ,MB ) Mg,n (M ), B B hence Kg,n (M) → Mg,n (M ) is quasiﬁnite and of ﬁnite type. Properness now follows from the valuative criterion, which is the content of the following Proposition 4.3. ♠ Proposition 4.3. Let R be a discrete valuation ring, with spectrum T having generic point η and special point t. Fix a morphism T → S and let Cη → M be an npointed twisted stable map over the generic point of T , and C → M a stable map over T extending the coarse map Cη → M . Then there is a discrete valuation ring R1 containing R as a local subring, and corresponding morphism of spectra T1 → T , and an npointed twisted stable map CT1 → M, such that the restriction CT1 η1 → M is isomorphic to the base change Cη ×T T1 → M, and the coarse map coincides with C ×T T1 → M . Such an extension, when it exists, is unique up to a unique isomorphism. Before proving this Proposition, we need to extend a few basic results known in case M is a Deligne–Mumford stack. Lemma 4.4 (Purity Lemma). Let M be a separated tame stack with coarse moduli space M . Let X be a separated scheme of dimension 2 satisfying Serre’s condition S2. Let P ⊂ X be a ﬁnite subset consisting of closed points, U = X P . Assume that the local fundamental groups of U around the points of P are trivial and that the local Picard groups of U around points of P are torsion free. Let f : X → M be a morphism. Suppose there is a lifting fU : U → M:
nn66 M nnn nn nnn nnn f nn
e fU U Then the lifting extends to X : // X // M U n66 == M nnn nnn f e nn nnn f nnn // //
e fU X M The lifting f is unique up to a unique isomorphism. Remark 4.5. We will use this lemma principally in the following cases: (1) X is regular, (2) X is normal crossings, locally Spec(R[x, y ]/xy )sh , with R regular, or (3) X = X0 × G, with X0 one of the ﬁrst two cases and G a linearly reductive ﬁnite group scheme. 18 ABRAMOVICH, OLSSON, AND VISTOLI Proof. The question is local in the ´tale topology, so we may replace X by e its strict henselization over some point p ∈ P , and correspondingly we may replace M and M by the strict henselization at f (p). By [4, Theorem 3.2 (d)]we can write M = [V /G], where V → M is a ﬁnite morphism and G → M a linearly reductive group scheme acting on V . By [4] Lemma 2.20 we have an exact sequence 1 −→ ∆ −→ G −→ H −→ 1 of group schemes over M , where ∆ is diagonalizable and H is tame and ´tale. e The morphism fU is equivalent to the data of a Gtorsor PU → U and a Gequivariant morphism U → V . We ﬁrst wish to extend PU over X . def Consider the H torsor QU = PU /∆ → U . Since the local fundamental groups of U around P are trivial, this H torsor is trivial, and extends trivially to an H torsor Q → X . Now PU → QU is a ∆torsor. We claim that it extends uniquely to a ∆torsor P → Q. Since ∆ is diagonalizable, it suﬃces to treat the case ∆ = µr . In this case PU → QU corresponds to an r torsion line bundle with a chosen trivialization of its r th power. The line bundle extends to X by the assumption on the local Picard groups, and the trivialization extends by the S2 assumption. Next we need to extend the principal action of G on PU to P . The closure Γ of the graph of the morphism G ×M PU → PU inside the scheme G ×M P ×X P is ﬁnite over G ×M P , and an isomorphism over G ×M PU . As G ×M P is ﬁnite and ﬂat over the S2 scheme X , it is also S2. It follows that the morphism Γ → G ×M P is an isomorphism, hence the action extends to a morphism G ×M P → P . It is easy to check that this is a principal action, as required. The same argument shows that the equivariant morphism PU → V extends to an equivariant morphism P → V , as required. ♠ Proof of Proposition 4.3. Step 1: We extend Cη → M over the generic points of the special ﬁber Cs . The question is local in the ´tale topology of C , therefore we can replace e C by its strict henselization at one of the generic points of Cs . We may similarly replace M by its strict henselization at the image point, and hence assume it is of the form [V /G], where V → M is ﬁnite and G a locally well split group scheme, see [4], Theorem 3.2 (d) and Lemma 2.20. We again write G as an extension 1 −→ ∆ −→ G −→ H −→ 1. The morphism Cη = Cη → M is equivalent to the data of a Gtorsor Pη → Cη and a Gequivariant morphism Pη → V . By Abhyankar’s lemma, the def H torsor Qη = Pη /∆ → Cη extends uniquely to Q → C after a base change on R. We want to extend the ∆torsor Pη → Qη to a ∆torsor P → Q. TWISTED YET TAME 19 Factoring ∆, we may assume ∆ = µr . In this case Pη → Qη corresponds to an r torsion line bundle with a chosen trivialization of its r th power. This line bundle is trivial since we have localized, hence it trivially extends to Q. The extension of the trivializing section may have a zero or pole along Cs , but after an r th order base change on R this zero or pole has multiplicity r · l for some integer l. Twisting the line bundle by OC (lCs ), the trivialization extends uniquely over Cs as required. We need to lift the principal action of G on P . Write again Γ for the closure of the graph of G ×T Pη → Pη inside G ×T Pη ×C Pη . This is a ∆equivariant subscheme, with respect to the action δ ·(g, p1 , p2 ) = (δg, p1 , δp2 ). The action is free, as it is already free on the last factor. Therefore the projection Γ → Γ/∆ is a ∆torsor. The quotient Γ/∆ includes the subscheme (Γ/∆)η = Qη . We claim that this is schemetheoretically dense in Γ/∆, because any nonzero sheaf of ideals with support over s would pull back to a nonzero sheaf of ideals on Γ. But Γη is by deﬁnition scheme theoretically dense in Γ. Since Q is normal, it follows that Γ/∆ = Q. Now as both Γ and P are ∆torsors over Q and Γ → P is ∆equivariant, this morphism is an isomorphism. Hence we have a morphism G × P → P , extending the action. It is again easy to see this is a principal action with quotient C . It is also clearly unique. Exactly the same argument shows that the morphism Pη → V extends uniquely to a Gequivariant P → V , as required. Step 2: We extend the twisted stable map C → M over the general locus Cgen , simply by applying the purity lemma (Lemma 4.4). Uniqueness in the Purity Lemma implies that the extension is unique up to unique isomorphism. Step 3: We extend the twisted curve C and the twisted stable map C → M along the smooth locus of C . Consider the index ri of Cη over a marking ΣC of C . Let Csm be the stack i obtained by taking ri th root of ΣC on C for all i. Then (Csm )η is uniquely i isomorphic to (Cη )sm . We need to construct a representable map Csm → M lifting Csm → M . The problem is local in the ´tale topology of Csm , so we may present Csm at a e point p on ΣC ∩ Cs as [D/µri ], where D is smooth over T , with unique ﬁxed i point q over p. Applying the purity lemma to the map D {q } → M, we have a unique extension D → M. The uniqueness applies also for µri × D → M, which implies that this object is µri equivariant, giving a unique morphism [D/µri ] → M, as needed. Step 4: We extend the twisted curve C and the twisted stable map C → M over the closure of the singular locus of Cη . This is similar to step 3. Step 5: Extension over isolated singular points. Consider an isolated node p of C . Passing to an extension on R we may assume it is rational over the residue ﬁeld k. Its strict henselization is isomorphic to (Spec R[u, v ]/(uv − π l ))sh , where π is a uniformizer of R, and 20 ABRAMOVICH, OLSSON, AND VISTOLI by Remark A.9 there is a twisted curve Cl which is regular over p with index l. This twisted curve has a chart of type [D/µl ], where the strict henselization of D looks like Spec R[x, y ]/(xy − π ), where µl acts via (x, y ) → (ζl x, ζl−1 y ), and u = xl , v = y l . There is a unique point q of D over p. To construct Cl → M it suﬃces to apply the purity lemma to the map D {q } → M; the resulting extension D → M is µl equivariant by applying the Purity Lemma to µl × D → M. We need to replace the morphism Cl → M, as it is not necessarily representable, and the construction of Cl does not commute with base change. Consider the morphism Cl → C × M. This morphism is proper and quasiﬁnite. Let Cl → C → C × M be the relative moduli space. By the local computation (Proposition 3.3), C is a twisted curve, hence C → M is a twisted stable map. Further, its formation commutes with further base change since the formation of moduli space does. ♠ 5. Reduction of Spaces of Galois admissible covers Let S be a scheme and G/S a locally wellsplit ﬁnite ﬂat group scheme. We can then consider the stack Kg,n (B G) of twisted stable maps from genus g twisted curves with n marked points to the classifying stack B G. This is the stack which to any S scheme T associates the groupoid of data (C , {Σi }n , h : C → B G), i=1 where (C , Σi }n is an nmarked twisted curve such that the coarse space i=1 (C, {pi }n ) (with the marking induced by the Σi ) is a stable npointed i=1 curve in the sense of DeligneMumfordKnudsen, and h is a representable morphism of stacks over S . The main result of this section is the following: Theorem 5.1. The stack Kg,n (B G) is ﬂat over S . To prove this theorem we in fact prove a stronger result. Let Mtw (G) denote the stack over S which associates to any S scheme g,n T the groupoid of data (C , {Σi }n , h : C → B G), i=1 where (C , {Σi }n ) is an nmarked twisted curve over T and h is a morphism i=1 of stacks over S (so (C, {pi }n ) is not required to be stable and h is not i=1 necessarily representable). Lemma 5.2. The natural inclusion Kg,n (B G) ⊂ Mtw (G) is representable g,n by open immersions. Proof. This is because the condition that an npointed nodal curve is stable is an open condition as is the condition that a morphism of stacks C → B G is representable, see Corollary C.7 and the discussion preceding it. ♠ TWISTED YET TAME 21 To prove 5.1 it therefore suﬃces to show that Mtw (G) is ﬂat over S . For g,n this we in fact prove an even stronger result. Let Mtw denote the stack g,n deﬁned in A.6. Forgetting the map to B G deﬁnes a morphism stacks (5.2.1) Mtw (G) → Mtw ×Spec(Z) S. g,n g,n Since Mtw is smooth over Z by A.6 the following theorem implies 5.1. g,n Theorem 5.3. The morphism 5.2.1 is a ﬂat morphism of algebraic stacks. Equivalently: let T be a scheme and C → T a twisted curve. Then HomT (C , BG) is ﬂat over T . The proof of 5.3 occupies the remainder of this section. Note ﬁrst of all that Mtw (G) is an algebraic stack locally of ﬁnite preg,n sentation over Mtw . Indeed the stack Mtw (G) is equal to the relative g,n g,n Homstack HomMtw (C univ , B G × Mtw ), g,n g,n where C univ → Mtw denotes the universal twisted curve. This stack is g,n algebraic locally of ﬁnite type over Mtw by [30, 1.1]. g,n We prove 5.3 by ﬁrst studying two special cases and then reducing the general case to these special cases. 5.4. The case when G is a tame ´tale group scheme. In this case we e claim that 5.2.1 in fact is ´tale. Indeed to verify this it suﬃces to show that e 5.2.1 is formally ´tale since it is a morphism locally of ﬁnite presentation. e If T0 ֒→ T is an inﬁnitesimal thickening and CT → T is a twisted curve over T , then the reduction functor from Gtorsors on CT to Gtorsors on CT0 is an equivalence of categories since G is ´tale. e 5.5. The case when G is locally diagonalizable. Let T be an S scheme, and let C → T be a twisted curve. The stack over T of morphisms C → BG is then equivalent to the stack TORSC /T (G) associating to any T ′ → T the groupoid of Gtorsors on C ×T T ′ . Let X denote the Cartier dual Hom(G, Gm ) so that G = SpecS OS [X ]. We use the theory of Picard stacks  see [7, XVIII.1.4] . Let PicC /T denote the Picard stack of line bundles on C , and let PicC /T denote the rigidiﬁcation of PicC /T with respect to Gm , so that PicC /T is the relative Picard functor of C /T . By Lemma 2.7 we have that PicC /T is an extension of a semiabelian group scheme by an ´tale group scheme. e Let PicC /T [X ] denote the group scheme of homomorphisms X → PicC /T . Lemma 5.6. The scheme PicC /T [X ] is ﬂat over T . Proof. The assertion is clearly local in the ´tale topology on S so we may e assume that S is connected and G diagonalizable. We may write G = µni and X = Z/(ni ). Then PicC /T [X ] = T PicC /T [Z/(ni )]. 22 ABRAMOVICH, OLSSON, AND VISTOLI It then suﬃces to consider the case where X = Z/(n), in which case ×n PicC /T [X ] = PicC /T [n]. This is the ﬁber of the map PicC /T −→ PicC /T over the identity section. This is a map of smooth schemes of the same dimension. It has ﬁnite ﬁbers since PicC /T is an extension of a semiabelian group scheme by an ´tale group scheme. Therefore this map is ﬂat, and e hence PicC /T [X ] → T is ﬂat as well. ♠ Let PicC /T [X ] denote the Picard stack of morphisms of Picard stacks X → PicC /T , where X is viewed as a discrete stack. Then PicC /T [X ] is a G–gerbe over PicC /T [X ], hence ﬂat over T . The result in the locally diagonalizable case therefore follows from the following lemma: Lemma 5.7. There is an equivalence of categories TORSC /T (G) → PicC /T [X ], were TORSC /T (G) denotes the stack associating to any T ′ → T the groupoid of Gtorsors on C ×T T ′ . Proof. For any morphism s : C → BG, the pushforward s∗ OC on C × BG has a natural action of G and therefore decomposes as a direct sum ⊕x∈X Lx . Base changing to C → C × BG (the map deﬁned by the trivial torsor) one sees that each Lx is a line bundle, and that the algebra structure on s∗ OC deﬁnes isomorphisms Lx ⊗ Lx′ → Lx+x′ giving a morphism of Picard stacks F : X → PicC /T [X ]. Conversely given such a morphism F , let Lx denote F (x). The isomorphisms F (x + x′ ) ≃ F (x) + F (x′ ) deﬁne an algebra structure on ⊕x∈X Lx . The relative spectrum over C × BG maps isomorphically to C and therefore deﬁnes a section. ♠ Remark 5.8. With a bit more work, one can prove a more general result which may be of interest: given a twisted curve C → T and a Ggerbe G → C , the stack SecT (G /C ) is a Ggerbe over its rigidiﬁcation SecT (G /C ), and the T space SecT (G /C ) is a pseudotorsor under the ﬂat groupscheme PicC /T [X ]. In particular SecT (G /C ) → T is ﬂat. TWISTED YET TAME 23 5.9. Observations on ﬁxed points. Before we consider general G, we make some observations about schemes of ﬁxed points of group actions on semiabelian group schemes. Let S be a scheme, and let A → S be a smooth abelian group scheme. Let Ao denote the connected component of the identity and assume we have an exact sequence of group schemes (5.9.1) 0 → Ao → A → W → 0, with W a ﬁnite ´tale group scheme over S . In what follows we will assume e that W is a constant group scheme (this always hold after making an ´tale e base change on S ) and that Ao is a semiabelian group scheme. Let H be a ﬁnite group of order invertible on S acting on A (by homomorphisms of group schemes). Let N denote the order of H . Since Ao is a semiabelian group scheme multiplication by N is surjective and ´tale on e Ao . It follows that multiplication by N on A is also ´tale and that there is e an exact sequence A − − → A − − → W/N W − − → 0. −− −− −− In particular, the image of ×N : A → A is an open and closed subgroup scheme A′ ⊂ A preserved by the H action. Lemma 5.10. The scheme of ﬁxed points AH is a smooth group scheme over S . Proof. We verify the inﬁnitesimal lifting property. Let T0 ֒→ T be a closed immersion deﬁned by a squarezero ideal I ⊂ OT , and let p0 ∈ AH (T0 ) be an H invariant point. We show that p0 can be lifted to an H invariant point of A(T ). Since A is smooth we can after perhaps shrinking on T lift p0 to some point p ∈ A(T ). Deﬁne q := ˜
h∈H ×N ph ∈ A(T ), where ph denotes the image of p under h : A → A. Note that in fact q ∈ AH (T ) and that q reduces to N p0 in A(T0 ) so that q ∈ A′H (T ). Since ˜ ˜ ˜ N is invertible in S multiplication by N : A → A′ is ´tale. Therefore after e replacing T by an ´tale extension there exists an element q ∈ A(T ) such e that N q = q . In particular, the reduction q0 ∈ A(T0 ) of q is an element with ˜ N (q0 − p0 ) = 0. Since the group scheme A[N ] is ´tale over T we can after e changing q by a point of A[N ] assume that q reduces to p0 . For any h ∈ H , the point q h − q ∈ A(T ) is a point annihilated by N since N (q h − q ) = (N q )h − (N q ) = q h − q = 0. ˜ ˜ Since the reduction map A[N ](T ) → A[N ](T0 ) is injective it follows that q ∈ AH (T ). ♠ 24 ABRAMOVICH, OLSSON, AND VISTOLI Now let X be a ﬁnitely generated Z/pn –module with H action, where p is prime to N . Let A[X ] denote the scheme of homomorphisms X → A. The group H acts on A[X ] as follows. An element h ∈ H sends a homomorphism ρ : X → A to the homomorphism X − − → X − − → A − − → A. −− −− −− H is deﬁned to be the ﬁber product The group scheme of ﬁxed points (A[X ]) of the diagram A[X ] ∆ (5.10.1) h −1 ρ h −−−−− A[X ] − − − − − → Q h∈H (haction) h∈H A[X ]. Note that the group scheme of ﬁxed points (A[X ])H is potentially quite unrelated to AH . Proposition 5.11. The group scheme of ﬁxed points (A[X ])H is ﬂat over S. Proof. Choose a presentation of X as an H representation (5.11.1) 0 → K → F → X → 0, where the underlying groups of K and F are free Z–modules of ﬁnite rank. Let A[F ] (resp. A[K ]) denote the space of homomorphisms F → A (resp. K → A). Applying Hom(F, −) (resp. Hom(K, −)) to 5.9.1 we see that A[F ] (resp. A[K ]) sits in a short exact sequence 0 → Ao [F ] → A[F ] → Hom(F, W ) → 0 (resp. 0 → Ao [K ] → A[K ] → Hom(K, W ) → 0), where Ao [F ] (resp. Ao [K ]) is a semiabelian group scheme. Note also that the relative dimension over S of A[F ] (resp. A[K ]) is equal to dim(A) · rk(F ) (resp. dim(A) · rk(K )). Since X is a torsion module rk(F ) = rk(K ) so A[F ] and A[K ] have the same dimension. The inclusion K ֒→ F induces a homomorphism (5.11.2) whose kernel is A[X ]. Lemma 5.12. The induced map Ao [F ] → Ao [K ] is surjective. Proof. The short exact sequence 5.11.1 induces an exact sequence of fppfsheaves on S Ao [F ] → Ao [K ] → Ext1 (X, Ao ). Since Ext1 (X, Ao ) = 0 (since Ao is divisible) the surjectivity of Ao [F ] → Ao [K ] follows. ♠ A[F ] → A[K ] TWISTED YET TAME 25 By 5.10 the induced morphism (5.12.1) f : (A[F ])H → (A[K ])H is a morphism of smooth group schemes with kernel the ﬁnite group scheme (A[X ])H (which is ﬁnite since A[X ] is a ﬁnite ﬂat group scheme). Note that the connected component of the identity in (A[F ])H (resp. (A[K ])H ) is equal to the connected component of the identity in Ao [F ]H (resp. Ao [K ]H ). Lemma 5.13. The morphism Ao [F ]H → Ao [K ]H induced by 5.12.1 is surjective. Proof. Let k be a ﬁeld and suppose given an H invariant homomorphism f : K → Ao over k. We wish to show that after possibly making a ﬁeld ˜ extension of k we can ﬁnd an extension f : F → Ao of f which is H invariant. Replacing S by Spec(k), for the rest of the proof we view everything as being over k. Consider the short exact sequence of abelian group schemes with H action 0 → Ao [X ] → Ao [F ] → Ao [K ] → 0. Viewing this sequence as an exact sequence of abelian sheaves on BHfppf we obtain by pushing forward to Spec(k)fppf an exact sequence Ao [F ]H (k) → Ao [K ]H (k) → H 1 (BHfppf , Ao [X ]). The lemma therefore follows from Lemma 5.14 below (which is stated in the generality needed later). ♠ Lemma 5.14. Let D be an abelian sheaf on BHT,fppf such that every local section of D is torsion of order prime to the order of H . Let f : BHT,fppf → Tfppf be the topos morphism deﬁned by the projection. Then Ri f∗ D = 0 for all i > 0. Proof. Let T• → BHT be the simplicial scheme over BHT associated to the covering T → BHT deﬁned by the trivial torsor (as in [24, 12.4]). For p ≥ 0 let fp : Tp,fppf → Tfppf be the projection. Note that Tp is a ﬁnite disjoint union of copies of T , and therefore for any abelian sheaf F on Tp,fppf we have Ri fp∗ F = 0 for i > 0. From this it follows that Rf∗ D is quasiisomorphic in the derived category of abelian sheaves on Tfppf to the complex f0∗ DT0 → f1∗ DT1 → · · · . Therefore the sheaf Ri f∗ D is isomorphic to the sheaf associated to the presheaf which to any T ′ → T associates the group cohomology (5.14.1) H i (H, D (T ′ )), where D (T ′ ) denotes the H module obtained by evaluating D on the object T ′ → BH corresponding to the trivial torsor on T ′ . Since D (T ′ ) is a direct 26 ABRAMOVICH, OLSSON, AND VISTOLI limit of groups of order prime to the order of H , it follows that the groups 5.14.1 are zero for all i > 0. ♠ We now complete the proof of Proposition 5.11. It follows that the map on connected components of the identity (A[F ]H )o → (A[K ]H )o is a surjective homomorphism of smooth group schemes of the same dimension, and hence a ﬂat morphism. Therefore the morphism A[F ]H → A[K ]H is also ﬂat and hence its ﬁber over the identity section, namely A[X ]H , is ﬂat over S . ♠ 5.15. General G: setup. We now return to the proof of 5.3. The assertion that 5.2.1 is ﬂat is clearly local in the ﬂat topology on S . We may therefore assume that G is wellsplit so that there is a split exact sequence 1 → D → G → H → 1, with H ´tale and tame and D diagonalizable. The map G → H induces a e morphism over Mtw × S g,n (5.15.1) Mtw (G) → Mtw (H ). g,n g,n We may apply the case of ´tale group scheme (Section 5.4) to the group e scheme H .. Therefore we know that Mtw (H ) is ﬂat (even ´tale) over Mtw × e g,n g,n S so it suﬃces to show that the morphism 5.15.1 is ﬂat. For this in turn it suﬃces to show that for any morphism t : T → Mtw (H ), with T the spectrum of an artinian local ring, the induced map g,n (5.15.2) Mtw (G) ×Mtw (H ) T → T g,n g,n is ﬂat. Let (C → T, F : C → BH ) be the npointed genus g twisted curve together with the morphism F : C → BH corresponding to t, and let C ′ → C denote the H torsor obtained by pulling back the tautological H torsor over BH . So we have a diagram with cartesian square C′ − − → C −− F T. For the rest of the argument we may replace S by T , pulling back all the objects over S to T . The action of H on D deﬁnes a group scheme D over BHT whose pullback along the projection T → BHT is the group scheme D with descent data T − − → BHT −− f TWISTED YET TAME 27 deﬁned by the action. Let D denote the pullback of this group scheme to C . The pullback of D to C ′ is equal to D ×T C ′ , but D is a twisted form of D over C . Similarly, the character group X of D has an action of the group H , giving a group scheme X over BHT which is Cartier dual to D . Let G → C denote the stack C ×BH BG, and let G ′ → C ′ denote the pullback.The stack G is a gerbe over C banded by the sheaf of groups D. To prove that the morphism 5.15.2 is ﬂat we need to show that the stack SecT (G /C ) is ﬂat over T . As T is local artinian, we may assume that SecT (G /C ) → T is set theoretically surjective, and replacing T by a ﬁnite ﬂat cover we may also assume that SecT (G /C ) → T has a section over the reduction of T . Note that the gerbe G ′ → C ′ is trivial: there is an isomorphism G ′ ≃ C ′ × BD. This is because T → BH corresponds to the trivial torsor, and any trivialization gives an isomorphism T ×BH BG ≃ BD. It follows that SecT (G ′ /C ′ ) = TORSC ′ /T (D ) as a scheme over T . However this isomorphism is not canonical. 5.16. Reduction to C ′ connected. We wish to apply previous results such as 5.7 to C ′ → T , but our discussion applied only when this is connected. We reduce to the connected case as follows: Suppose the H bundle C ′ → C has disconnected ﬁber over the residue ﬁeld of T , then since T is artinian we can choose a connected component C ′′ ⊂ C ′ . Let H ′′ ⊂ H be the subgroup sending C ′′ to itself. Then C ′′ − > C is an H ′′ bundle. If we denote G′′ = G ×H H ′′ , there is an equivalence of categories between Gbundles E → C lifting the given H bundle C ′ and G′′ bundles E ′′ → C lifting the H ” bundle C ′′ : one direction is by restricting E to C ′′ ⊂ C ′ , the other direction by inducing E ′′ → C from G′′ to G. It therefore suﬃces to consider the case when C ′ → T has connected ﬁbers. 5.17. General G: strategy. Our approach goes as follows: (1) By Lemma 5.7 we have a precise structure, as a gerbe over a group scheme, of the stack SecT (G ′ /C ′ ). (2) This provides an analogous precise stucture on SecBHT (G /C ) and its rigidiﬁcation SecBHT (G /C ): we have that SecBHT (G /C ) is a D gerbe over SecBHT (G /C ). However SecBHT (G /C ) is not a groupscheme but a torsor (see below). (3) The stack SecT (G /C ) = SecT SecBHT (G /C ) / BHT can be thought of as a stack theoretic version of pushforward in the fppf topology from BHT to T . Indeed, for a representable morphism U → BHT , the fppf sheaf associated to the space SecT (U/BHT ) coincides with f∗ (U )f ppf . We analyze the structure of the pushforward of its building blocks, namely the rigidiﬁcation SecBHT (G /C ) and the groupschemes underlying the torsor and gerbe structures. 28 ABRAMOVICH, OLSSON, AND VISTOLI 5.18. Structure of SecT (G ′ /C ′ ) and SecBHT (G /C ). Let us explicitly state the structure in (1) above: • SecT (G ′ /C ′ ) → SecT (G ′ /C ′ ) is a gerbe banded by the groupscheme D , and • the rigidiﬁcation SecT (G ′ /C ′ ) of SecT (G ′ /C ′ ) is isomorphic to the T group scheme PicC ′ /T [X ]. We proceed with the structure in (2). Lemma 5.19. The automorphism group of any object of SecBHT (G /C ) over Z → BH is canonically isomorphic to D (Z ) Proof. This follows from the facts that G is a D gerbe and C → BH connected. Indeed if s : C → G is a section, then an automorphism of s is given by a map C → D . This necessarily factors through BHT since by connectedness F∗ OC = OBHT (where F : C → BHT is the structure morphism) and D is aﬃne over BHT . ♠ We can deﬁne SecBHT (G /C ) to be the rigidiﬁcation, (which is also the relative coarse moduli space since SecBHT (G /C ) → BH is representable). We have that SecBHT (G /C ) → SecBHT (G /C ) is a gerbe banded by D . We also have naturally that SecT (G ′ /C ′ ) = T ×BHT SecBHT (G /C ). The lemma above and the base change property of rigidiﬁcation gives SecT (G ′ /C ′ ) = T ×BHT SecBHT (G /C ). Let us look at the structure underlying SecBHT (G /C ). The group H acts on PicC ′ /T and on X . We therefore also obtain a left action of H on PicC ′ /T [X ] = Hom(X, PicC ′ /T ), where h ∈ H sends a homomorphism ρ : X → PicC ′ /T to the homomorphism (5.19.1) X − − → X − − → PicC ′ /T − − → PicC ′ /T . −− −− −−
h −1 ρ h This deﬁnes an fppf sheaf on BHT , which is easily seen to be represented by PicC /BHT [X]. Note that in general the identiﬁcation SecT (G ′ /C ′ ) = T ×BHT SecBHT (G /C ) does not give an isomorphism of SecBHT (G /C ) with PicC /BHT [X], because the descent data may in general be diﬀerent. However, we have the following: Lemma 5.20. There is a natural action of PicC /BHT [X] on SecBHT (G /C ), which pulls back to the natural action of PicC ′ /T [X ] on itself by translation. In particular SecBHT (G /C ) is a torsor under PicC /BHT [X]. Proof. Let Φ : X → PicC /BHT be an object of PicC /BHT [X] over some arrow ξ : Z → BHT and s : C → G a section, again over Z . Then we can deﬁne a new section sΦ : C → G as follows. Let P Φ denote the D torsor corresponding to Φ (see Lemma 5.7). The section s deﬁnes an isomorphism (which we denote by the same letter) s : C × BD → G, TWISTED YET TAME 29 and we let sΦ denote the composite C − − → C × BD − − → G. −− −− By the universal property of rigidiﬁcation this deﬁnes an action of PicC /BHT [X] on SecBHT (G /C ). The fact that its pullback is identiﬁed with the action of PicC ′ /T [X ] on itself by translation is routine. ♠ 5.21. The pushforward of PicC /BHT [X] and its torsor SecBHT (G /C ). We write (PicC /BHT [X])fppf for the sheaf in the fppf topology on BHT represented by PicC /BHT [X]. The fppf pushforward f∗ ((PicC /BHT [X])fppf ) is represented by the scheme of ﬁxed points PicC ′ /T [X ]H . We have: Lemma 5.22. The scheme of ﬁxed points PicC ′ /T [X ]H is ﬁnite and ﬂat over T. Proof. Note that in the notation of Section 2.11, since X is a torsion group we have PicC ′ /T [X ] = Pico ′ /T [X ] C where Pico ′ /T is the group scheme of degree0 line bundles deﬁned in 2.10. C Now by 2.11, we have that Pico ′ /T is an extension of a semiabelian group C scheme by an ´tale group scheme. The latter ´tale group scheme is ﬁnite e e since T is assumed Artinian local. The lemma therefore follows from Proposition 5.11. ♠ Now consider the torsor SecBHT (G /C ) and the pushforward under f of the corresponding sheaf (SecBHT (G /C ))fppf . We have the following. Lemma 5.23. Let E be an abelian sheaf on BHT,fppf such that every local section is torsion of order prime to the order of H , and let P → BHT be a E –torsor. Then the sheaf of sets f∗ P on Tfppf is a torsor under the sheaf f∗ E . Proof. The fact that P is a torsor under E immediately implies that f∗ P is a pseudotorsor under f∗ E . So the only issue is to show that fppf locally on T the sheaf f∗ P has a section. Equivalently, we need to show that after making a ﬂat surjective base change T ′ → T the torsor P itself becomes trivial. The class of the torsor is a class in H 1 (BHT , E ). By 5.14 and the Leray spectral sequence this lies in H 1 (T, f∗ E ). Any class in this group can be killed by making a ﬂat surjective base change T ′ → T . ♠ In our situation, with E = (PicC /BHT [X])fppf and P = SecBHT (G /C )fppf , we obtain that the sheaf f∗ (SecBHT (G /C ))fppf is represented by a torsor under PicC ′ /T [X ]H . In particular the space representing this sheaf is ﬂat over T . We denote this T space by the shorthand notation SecH  a complete notation would look like SecT (SecBHT (G /C )/BHT ). We turn our view to SecT (G /C ).
1×P Φ s 30 ABRAMOVICH, OLSSON, AND VISTOLI Lemma 5.24. The automorphism groupscheme of an object of SecT (G /C ) ober a T scheme B is the group scheme DH representing f∗ D . Proof. Let s : C → G be a section over some T scheme B . Since SecT (G /C ) = SecT (SecBH (G /C ) / BH ), an automorphism of s is a section over B × BH of the automorphism groupscheme of s viewed as an object of SecBH (G /C ) over B × BH . By Lemma 5.19 this is a section over B × BH of the group scheme D , namely a section of f∗ D , as required. ♠ We can deﬁne SecT (G /C ) to be the rigidiﬁcation. We have that SecT (G /C ) → SecT (G /C ) is a gerbe banded by the group scheme DH representing f∗ D . It therefore suﬃces to show that SecT (G /C ) is ﬂat. There is a natural map SecT (G /C ) → SecH inducing SecT (G /C ) → SecH . The following clearly suﬃces: Proposition 5.25. The morphism SecT (G /C ) → SecH is a gerbe banded by the group scheme D H representing f∗ D , and hence SecT (G /C ) → SecH is an isomorphism. Consider a morphism Z → SecH and the ﬁber product S := Z ×SecH Sec(G /C ). Lemma 5.26. Locally in the fppf topology on Z there exists a section Z → S. Proof. Let s ∈ f∗ (SecBHT (G /C ) (Z ) denote the section deﬁned by Z → SecH . ¯ By deﬁnition it corresponds to a morphism BHZ → SecBHT (G /C ) over the natural morphism BHZ → BHT . Since SecBHT (G /C ) is a D gerbe over SecBHT (G /C ), the obstruction to ﬁnding a section of S over s is equal to the ¯ class of the D gerbe of liftings of s : BHZ → SecBHT (G /C ) to a morphism ¯ 2 BHZ → SecBHT (G /C ). This class lies in Hfppf (BHZ , D ). Since by Lemma 5.14 we have Ri f∗ D = 0 for i > 0, the spectral sequence puts this class in 2 Hfppf (Z, f∗ D ). This vanishes when pulled back to an fppf cover, and the lemma follows. ♠ If s, s′ ∈ S(Z ) are two sections, then we claim that after making a ﬂat base change on Z the sections s and s′ are isomorphic. Indeed let I be the sheaf over BHZ,fppf of isomorphisms between the two morphisms BHZ → SecBHT (G /C ) corresponding to s and s′ . Then I is a D torsor over BHZ,fppf, and since R1 f∗ D = 0 it follows that fppf locally on Z this torsor is trivial. It follows that S is a gerbe banded by f∗ D . This completes the proof of 5.25, implying 5.3. ♠ 6. Example: reduction of X(2) in characteristic 2 6.1. X(2) as a distinguished component in K0,4 (B µ2 ). Let X(2) denote the stack over Q associating to any scheme T the groupoid of pairs (E, ι), where E/T is a generalized elliptic curve with a full level 2structure ι : (Z/2)2 ≃ E [2] in the sense of DeligneRapoport [12]. TWISTED YET TAME 31 Order the elements of (Z/2)2 in some way. For any pair (E, ι) ∈ X(2)(T ), the involution P → −P has ﬁxed points the 2torsion points E [2] ⊂ E , and hence the stacktheoretic quotient [E/ ± 1] of E by this involution comes equipped with an ordered set of gerbes Σi ⊂ [E/ ± 1] in the smooth locus. Furthermore, one checks by direct calculation on the geometric ﬁbers of E that the coarse space of [E/ ± 1] with the resulting four marked points is a stable genus 0 curve with four marked points. In this way we obtain a morphism of stacks X(2) → K0,4 (B µ2 )Q sending (E, ι) → (E → [E/ ± 1], {Σi ⊂ [E/ ± 1]}). One veriﬁes immediately that this functor is fully faithful, and identiﬁes X(2) with the closed substack KQ ⊂ K0,4 (B µ2 )Q classifying data (B → P, {Σi }) where B → P is a µ2 torsor such that the resulting map B → P to the coarse space of P is ramiﬁed over each of the marked points of P . We have seen that K0,4 (B µ2 ) is ﬂat over Spec(Z). We want to have a nice description of the closure K of KQ , preferably as a naturally deﬁned moduli stack ﬂat over Z. It turns out that this can be done in the best possible way: K is an open and closed substack of K0,4 (B µ2 ), deﬁned naturally in terms of the behavior of the µ2 cover at the marked points. One can describe this directly, but we ﬁnd that this gives us a good pretext for introducing the rigidiﬁed cyclotomic inertia stack and evaluation maps. 6.2. Cyclotomic inertia. Recall that in [3, §3] one deﬁned the cyclotomic inertia stacks: for ﬁxed positive integer r ≥ 1 we denote by Iµr (X ) → X the stack whose objects are pairs (ξ, φ) where ξ is an object of X and φ : µr → Aut(ξ ) is a monomorphism, and whose arrows are commutative diagrams as usual; and
∞ Iµ(X ) =
r =1 Iµr (X ). For a noetherian stack with ﬁnite diagonal, The argument of [3, Proposition 3.1.2] shows that the morphism Iµ(X ) → X is representable by quasiprojective schemes. Indeed, the ﬁber over an object ξ ∈ X (T ) is the scheme r Homgr−sch/T (µr , AutT (ξ )), which is quasiprojective as the relevant r is bounded. Proposition 6.3. If X is tame, the morphism Iµ(X ) → X is ﬁnite. Proof. As the problem is local in the fppf topology of the coarse moduli space X of X , we may assume X = [U/G] with G a linearly reductive groupscheme [4, Theorem 3.2 (c)]. The pullback IX ×X U is a closed subgroup scheme of the ﬁnite ﬂat linearly reductive group scheme GU . It therefore suﬃces to show the following Lemma. ♠ 32 ABRAMOVICH, OLSSON, AND VISTOLI Lemma 6.4. Let G1 and G2 be two ﬁnite ﬂat linearly reductive group schemes over a scheme U . The scheme Homgroupschemes /U (G1 , G2 ) is ﬁnite over U . Proof. The condition that a morphism G1,T → G2,T over some T is a group homomorphism is clearly a closed condition. It follows that the scheme Homgroupschemes /U (G1 , G2 ) is a disjoint union of quasiprojective schemes being a locally closed subscheme of the Hilbert scheme of G1 × G2 . To prove that Homgroupschemes /U (G1 , G2 ) is ﬁnite over U , we verify the valuative criterion for properness. The fact that Homgroupschemes /U (G1 , G2 ) → U is quasiﬁnite can be easily veriﬁed along similar lines. So suppose U is the spectrum of a discrete valuation ring V with generic point η = Spec(K ), and let fη : G1,η → G2,η be a homomorphism. Let p be the residue characteristic of V (p = 0 is allowed), and let ∆i ⊂ Gi be the subgroup scheme of ptorsion elements. Since the group schemes Gi are linearly reductive, the group schemes ∆i are locally diagonalizable and the quotients Hi = Gi /∆i are ´tale. The map fη restricts to a homomorphism e ¯ hη : ∆1,η → ∆2,η and induces a homomorphism fη : H1,η → H2,η . Since the group schemes ∆i are diagonalizable the morphism hη extends to a homomorphism h : ∆1 → ∆2 . Similarly since the group schemes Hi are ¯ ¯ ´tale the morphism fη extends to a homomorphism f : H1 → H2 . We then e need to ﬁnd a dotted arrow f ﬁlling in the diagram // ∆1 // G1 // H1 // 0 0
h f ¯ f 0 // ∆2 // G2 // H2 // 0 and restricting to fη on the generic ﬁber. The morphism f is clearly unique if it exists, so it suﬃces to prove its existence after replacing V by a ﬁnite ﬂat extension. By [4, 2.16] we may therefore assume that G1 = H1 ⋉ ∆1 is wellsplit. Since H is constant and G2 proper, the composite H1,η ⊂ G1,η
fη // G2,η then extends uniquely to a morphism of schemes g : H1 → G2 . Deﬁne a morphism of schemes f : G1 = H1 ⋉ ∆1 → G2 , (z, δ) → g(z ) · h(δ). This morphism is a homomorphism since it restricts to a homomorphism on the generic ﬁber. This completes the veriﬁcation of the valuative criterion and therefore the proof of that Homgroupschemes/U (G1 , G2 ) is ﬁnite over U . ♠ Remark 6.5. In fact the scheme Homgroupschemes/U (G1 , G2 ) in 6.4 is also ﬂat over U . TWISTED YET TAME 33 To prove this it suﬃces to consider the case when U is the spectrum of an artinian local ring A. After replacing A by an fppf cover we may assume that the group schemes Gi = Hi ⋉ ∆i are wellsplit, where ∆i is a local group scheme over U and Hi is ´tale. We then obtain a morphism e Homgroupschemes /U (G1 , G2 ) Homgroupschemes /U (∆1 , ∆2 ) × Homgroupschemes /U (H1 , H2 ). The target of this morphism is a product of ´tale group schemes of ﬁnite e type, so it suﬃces to show that this morphism is ﬂat. So ﬁx homomorphisms ¯ ˜ h : ∆1 → ∆2 and f : H1 → H2 . For any lifting f : H1 → G2 we then obtain a morphism of schemes ˜ f : H1 ⋉ ∆1 → H2 ⋉ ∆2 , (z, δ) → f (z )h(δ). A calculation shows that the condition for f to be a homomorphism of groupschemes is equivalent to the condition that the map h : ∆1 → ∆2 is compatible with the H1 actions, where H1 acts on ∆2 through the homo¯ morphism f : H1 → H2 . If Xi is the character group of ∆i and h∗ : X2 → X1 is the map deﬁning h, then h is compatible with the H1 actions if and only if h∗ is compatible with the H1 actions. This is clearly an open condition. It follows that to complete the proof of the claim it suﬃces to show that the ¯ ˜ scheme of liftings of f to a homomorphism f : H1 → G2 is ﬂat over U . ˜ is given by a map ξ : H1 → ∆2 (U ) deﬁned For this note that any lifting f by ˜ ¯ f (z ) = (f (z ), ξz ) ∈ H2 ⋉ ∆2 . If z, z ′ ∈ H1 are two elements, we then must have ˜ ¯ (f (zz ′ ), ξzz ′ ) = f (zz ′ ) ˜˜ = f (z )f (z ′ ) ¯ ¯ = (f (z ), ξz )(f (z ′ ), ξz ′ ) ′ ¯¯ = (f (z )f (z ′ ), ξ z + ξz ′ ).
z + ξz ′ , in other words ξ : H1 → ∆2 (U ) deﬁnes a 1cocycle for H1 So ξzz ′ = with coeﬃcients in ∆2 (U ). Conversely any 1cocycle for H1 with coeﬃcients ˜ in ∆2 (U ) deﬁnes a lifting f . Since the order of H1 is prime to the order of ∆2 (U ) such a 1cocycle is trivial in cohomology, so there exists an element m ∈ ∆2 (U ) such that ξz = mz − m in ∆2 (U ). This element m is uniquely determined up to adding an H1 invariant element of ∆2 (U ). From this it follows that the scheme of ¯ liftings of f is isomorphic to the diagonalizable group scheme deﬁned by the kernel of the homomorphism X2 → X2,H1 (where X2,H1 denotes the coinvariants). z′ ξz 34 ABRAMOVICH, OLSSON, AND VISTOLI 6.6. Rigidiﬁed cyclotomic inertia. Each object of Iµr (X ) has µr sitting in the center of its automorphisms. We can thus rigidify it and obtain a stack Iµr (X ) with a canonical morphism Iµr (X ) → Iµr (X ). Taking the disjoint union over r ≥ 1 we obtain Iµ(X ) → Iµ(X ). The stack Iµ(X ) is called the rigidiﬁed cyclotomic inertia stack of X . The stacks Iµr (X ) and Iµ(X ) have another interpretation, discussed in [3, Section 3.3]. Consider the 2category whose objects consist of morphisms of stacks α : G → X , where G is a gerbe banded by µr and α is a representable morphism; morphisms and 2arrows are deﬁned in the usual way. Since α is representable, this 2category is equivalent to a category. This is isomorphic to the stack Iµr (X ). This gives the stack Iµ(X ) the interpretation as the stack of cyclotomic gerbes in X , and Iµ(X ) → Iµ(X ) together with Iµ(X ) → X is the universal gerbe in X . The nonidentity components of Iµ(X ) are known as twisted sectors. 6.7. Evaluation maps. Consider now the stack of twisted stable maps Kg,n (X , β ). This comes with a diagram // Cg,n (X , β ) Σi J JJ JJ JJ JJ J$$ Kg,n (X , β ). // X where Cg,n (X , β ) → X is the universal twisted stable map and Σi are the n markings. For each i, the resulting diagram Σi // X Kg,n (X , β ).
is a cyclotomic gerbe in X parametrized by Kg,n (X , β ). This gives rise to n evaluation maps ei : Kg,n (X , β ) −→ Iµ(X ). Evaluation maps (and their sisters, twisted evaluation maps) are of central importance in Gromov–Witten theory of stacks, because the natural gluing maps Kg1 ,n1 +1 (X , β1 ) ×Iµ (X ) Kg2 ,1+n2 (X , β2 ) // Kg1 +g2 ,n1 +n2 (X , β2 ) have the ﬁbered product based on the evaluation map on the right and the twisted evaluation map, namely the evaluation map composed with inversion on the band, on the left. TWISTED YET TAME 35 6.8. Back to K ⊂ K0,4 (B µ2 ). The rigidiﬁed cyclotomic inertia stack of B µ2 has two components  the identity component is B µ2 itself, and the nonidentity component  the twisted sector  is a copy of the base scheme Spec Z. Since KQ corresponds to totally ramiﬁed maps, it is precisely the locus in K0,4 (B µ2 ) where all four evaluation maps land in the twisted sector. Since Iµ(B µ2 ) is ﬁnite unramiﬁed, the inverse image of each component is open and closed. Therefore K, the closure of KQ in K0,4 (B µ2 ) is open and closed. In particular it is ﬂat over Z. Also, since the generic ﬁber is irreducible, K is irreducible. It is simply the gerbe banded by µ2 over the λ line M0,4 ≃ P1 , 2 associated to the class in Hfppf (P1 , µ2 ) associated to c1 (OP1 (1). 6.9. What does KF2 parametrize? There is one little problem with the closure K of X (2)Q : in characteristic 2 it has nothing to do with elliptic curves. Given a smooth rational curves with 4 marked points, say at 0, 1, ∞ and λ, there is a unique µ2 bundle over the twisted curve which is degenerate over the coarse curve at all the markings, namely the scheme given in aﬃne coordinates by the equation y 2 = x(x − 1)(x − λ). In characteristic 2, this is a cuspidal curve of geometric genus 0, with its cusp given by x2 = λ. This well known example is possibly the simplest example of geometric interest of a principal bundle over a smooth scheme with singular total space. It is not diﬃcult to see that a similar situation holds at the node of a singular curve of genus 0: the singularity of the µ2 bundle is a tacnode. 6.10. X(2) as a distinguished component in K1,1 (B µ2 ). Let 2
◦ K1,1 (B µ2 ) ⊂ K1,1 (B µ2 ) 2 2 be the open substack classifying µ2 torsors over smooth elliptic curves. For 2 ◦ any object (E/T, P → E ) ∈ K1,1 (B µ2 ) over some scheme T , there is a 2 ◦ natural action of µ2 on this object given by the action on P . Let K1,1 (B µ2 ) 2 2 be the rigidiﬁcation. ◦ A simple, but not concrete, description of K1,1 (B µ2 ) was given in Lemma 2 5.7: let E/M1,1 be the universal elliptic curve. Then as discussed before, ◦ K1,1 (B µ2 ) = PicE/M1,1 [X ] where X = (Z/(2))2 . It follows that 2 K1,1 (B µ2 ) = PicE/M1,1 [2] ×M1,1 PicE/M1,1 [2]. 2 The compactiﬁcation over the boundary of M1,1 is not hard to describe as well. But we wish to describe this stack in terms of more classical objects. ◦ The stack K1,1 (B µ2 ) can be reinterpreted as follows. 2 ◦ Given an object (E/T, π : P → E ) ∈ K1,1 (B µ2 ) over a scheme T , the 2 sheaf π∗ OP on E has a µ2 action given by the action on P , and therefore 2
◦ 36 ABRAMOVICH, OLSSON, AND VISTOLI decomposes as a direct sum π∗ OP = ⊕χ∈(Z/2)2 Lχ . Furthermore, since P is a torsor over E each Lχ is a locally free sheaf of rank 1 on E . The sheaves Lχ therefore deﬁne a homomorphism φP : (Z/2)2 → Pic0 (E ) ≃ E, χ → [Lχ ]. Let H(2) denote the stack over Z associating to any scheme T the groupoid of pairs (E, φ), where E/T is an elliptic curve and φ : (Z/2)2 → E [2] is a homomorphism of group schemes over T . The above construction gives a functor ◦ K1,1 (B µ2 ) → H(2) 2 and a straightforward veriﬁcation shows that this functor induces an isomorphism ◦ K1,1 (B µ2 ) ≃ H(2). 2 For any quotient of ﬁnite abelian groups π : (Z/2)2 → A, let HA (2) denote the stack associating to any scheme T the groupoid of pairs (E, φA ), where E/T is an elliptic curve and φA : A → E [2] is an Astructure in the sense of KatzMazur [20, 1.5.1]. Sending such a pair (E, φA ) to (E, (Z/2)2 deﬁnes a morphism of stacks (6.10.1) HA (2) → H(2). Using [20, 1.6.2] one sees that this in fact is a closed immersion. We can describe the above in more classical notation as follows. Deﬁne stacks Y(2), Y1 (2), and Y(1) by associating to any scheme T the following groupoids: Y(2) : The groupoid of pairs (E, φ), where E/T is an elliptic curve and φ : (Z/2)2 → E [2] is a (Z/2)2 generator; Y1 (2) : The groupoid of pairs (E, φ), where E/T is an elliptic curve and φ : (Z/2) → E [2] is a (Z/2)generator; Y(1) : The groupoid of elliptic curves over T . Then 6.10.1 gives a closed immersion i(2) : Y(2) ֒→ H(2) corresponding to the identity map (Z/2)2 → (Z/2)2 , three closed immersions iK : Y1 (2) ֒→ H(2), j = 1, 2, 3, corresponding to the three surjections (Z/2)2 → Z/2 (indexed by index 2 subgroups K ⊂ (Z/2)2 ), and a closed immersion i(0) : Y(1) ֒→ H(2)
(1) π // A φA // E [2] ) TWISTED YET TAME 37 corresponding to the unique surjection (Z/2)2 → 0. The resulting map (6.10.2) Y(2) Y1 (2) Y1 (2) Y1 (2) Y(1) → H(2) is then a proper surjection, which over Z[1/2] is an isomorphism. Note in particular that the forgetful map H(2) → Y(1) sending (E, φ) to E has degree 16, as H(2) → Y(1) is ﬂat by 5.3 and over Z[1/2] the map clearly has degree 16. Over F2 , however, the map 6.10.2 joins together the various components in an interesting way. If E/k is an ordinary elliptic curve over a ﬁeld k of characteristic 2, then any homomorphism (6.10.3) φ : (Z/2)2 → E [2] ≃ Z/(2) × µ2 has a nontrivial kernel, and the irreducible components of Y(2)F2 are indexed by these kernels. Since a Z/(2)2 structure is surjective on geometric points, the kernel cannot be the whole group. Therefore Y(2)F2 has 3 irreducible components: for a subgroup K ⊂ (Z/2)2 of index 2 we get an irreducible component Y(2)K corresponding to pairs (E, φ), where the map 6.10.3 has F2 kernel K . Note also that for any ordinary elliptic curve E/k over a ﬁeld k, the ﬁber product of the diagram Y(2)K F2 Spec(k) − − → Y(1)F2 −− has length 2 over k. Similarly the reduction Y1 (2)F2 has two irreducible components Y1 (2)F2 = Y1 (2)′ ∪ Y1 (2)∅ , where Y1 (2)′ (resp. Y1 (2)∅ ) classiﬁes pairs (E, φ : Z/2 → E [2]) where the map φ is injective (resp. the zero map). This time only the ﬁber of Y1 (2)′ over a kpoint of Y(1)F2 has length 2. The closed ﬁber H(2)F2 then has four irreducible components. For a subgroup K ⊂ (Z/2)2 of index 2, we have an irreducible component ZK which is settheoretically identiﬁed with Y(2)K as well as the component Y1 (2)′ F2 given by the inclusion iK . The fourth component Z∅ is settheoretically identiﬁed with Y(1)F2 via i(0) , and with all the components Y1 (2)∅ via the (1) inclusions iK . Over the point of Y(1)(F2 ) given by the supersingular elliptic curve E , there is only one homomorphism (the zero map) φ : (Z/2)2 → E [2], and hence the four irreducible components of H(2)F2 all intersect at this point (and nowhere else). Note also that over the ordinary locus of Y(1), each of the components ZK has length 4 over Y(1)F2 as does the component Z∅ (so H(2)F2 has length 16 over Y(1)F2 , as we already knew).
(1) E 38 ABRAMOVICH, OLSSON, AND VISTOLI The reduced ﬁber over F2 looks roughly like this: (1,1)
(0,1) (1,0) φ=0 The following ﬁgure schematically describes the main component Y(2): 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 Here is the component Y1 (2)K for K generated by (1, 1): TWISTED YET TAME
11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 39 There are of course two more corresponding to the two other K of order 2. In 000000000000000000000 addition of course there is Y(1): 111111111111111111111
111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 The way the main component Y(2) meets Y1 (2)K is described as follows: 40 ABRAMOVICH, OLSSON, AND VISTOLI
1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 111111111 000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 and the other two are similar, of course along the corresponding component Y(2)K . F2 The main component Y(2) meets Y(1) only at the supersingular point, but Y1 (2)K meets Y(1) along the component corresponding to φ = 0:
11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 The complete picture is something like this: TWISTED YET TAME
1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 1111111111111111 0000000000000000 1111111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000000 11111111111111111111111111111111111111111 00000000000000000000000000000000000000000 1111111111111111 0000000000000000 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000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 111111111111111111111 000000000000000000000 41 An important question is that of describing explicitly the “most important” part of Kg,n (B µ2g ), in this example X(2). In genus 1, Katz and Mazur m gave an ideal solution: the moduli stack of generalized elliptic curves with (Z/m)2 structure, which is a closed substack of our K1,1 (B µ2 ), is regular m and ﬂat over M1,1 . But for higher genus a satisfactory conclusion is missing: in [20, Section 1.9] Katz and Mazur proposed a solution using norms, but it was shown to be ill behaved in [9, Appendix A]. It still may be of interest (m) to study the closure of the Qstack Mg of level m curves in Kg (B µ2g ). m Appendix A. Twisted curves and log curves by Martin Olsson In [29] we introduced a notion of log twisted curve, and proved that the category of log twisted curves (over some scheme S ) is naturally equivalent to the category of twisted DeligneMumford curves over S . The deﬁnition of a log twisted curve in loc. cit. included various assumptions about certain integers being invertible on the base scheme. In this appendix we show that if we remove these assumptions in the deﬁnition of log twisted curve, we obtain a category equivalent to the category of twisted curves. Deﬁnition A.1 ([25, 3.1]). Let X be an Artin stack. (i) A ﬁne log structure M on X is called locally free if for every geometric ∗ point x → X the monoid Mx := Mx /OX,x is isomorphic to Nr for some r . ¯ ¯ ¯ ¯ (ii) A morphism M → N of locally free log structures on X is called simple if for every geometric point x → X the monoids Mx and N x have ¯ ¯ ¯ the same rank, the morphism ϕ : Mx → N x is injective, and for every ¯ 42 ABRAMOVICH, OLSSON, AND VISTOLI irreducible element f ∈ N x there exists an irreducible element g ∈ Mx and ¯ ¯ a positive integer n such that ϕ(g) = nf . Remark A.2. This diﬀers from [29, 1.5] where the integer n in (A.1 (ii)) was assumed invertible in k(¯). x Let f : C → S be a nodal curve over a scheme S . As discussed in [29, §3], there exist canonical log structures MC and MS on C and S respectively, and an extension of f to a log smooth morphism (C, MC ) → (S, MS ). Deﬁnition A.3. A npointed log twisted curve over a scheme S is a collection of data (C/S, {σi , ai }n , ℓ : MS ֒→ M′ ), i=1 S where C/S is a nodal curve, σi : S → C are sections, the ai are positive integervalued locally constant functions on S , and ℓ : MS ֒→ M′ is a S simple morphism of log structures on S , where MS denotes the canonical log structure on S mentioned above. Let (C/S, {σi , ai }, ℓ : MS ֒→ M′ ) be a log twisted curve over S , and let S (A.3.1) (C, MC ) −→ (S, MS ) (C/S, {σi }) as in [29, 3.10] (note that MC is not equal to MC mentioned above, as MC also takes into account the marked points). We construct a twisted n–pointed curve (C , {Σi })/S from the log twisted curve (C/S, {σi , ai }, ℓ : MS ֒→ M′ ) as in [29, §4]. S Deﬁne C to be the ﬁbered category over S which to any h : T → S associated the groupoid of data consisting of a morphism s : T → C over h together with a commutative diagram of locally free log structures on T h∗ MS − − → h∗ M′ −− S τ s∗ MC − − → M′ , −− C
k ℓ be the morphism of log schemes obtained from the pointed curve (A.3.2) where: ¯ (1.3 (i)) The map k is a simple, and for every geometric point t → T , the ′ ′ map MS,t → MC,t is either an isomorphism, or of the form Nr → Nr+1 ¯ ¯ sending ei to ei for i < r and er to either er or er + er+1 . ¯ (1.3 (ii)) For every i and geometric point t → T with image under s in σi (S ) ⊂ C , the group (A.3.3) Coker(MS,t ⊕ MC,t −→ MC,t ) ¯ ¯ ¯
′gp gp ′gp ¯ is a cyclic group of order ai (t). TWISTED YET TAME 43 For every i, deﬁne Σi ⊂ C to be the substack classifying morphisms s : T → C which factor through σi (S ) ⊂ C and diagrams (A.3.2) such that for ¯ every geometric point t → T the image of (M′ t − τ (h∗ M′ t )) → OT,t ¯ C,¯ S,¯ is zero. Proposition A.4. The data (C , {Σi }n ) is a twisted npointed curve. i=1 Proof. This follows from the argument given in [29, §4]. The main result of this appendix is the following: Theorem A.5. Let S be a scheme. The functor (A.5.1) (npointed log twisted curves) → (twisted npointed curves) sending (C/S, {σi , ai }, ℓ : MS ֒→ M′ ) to the twisted npointed curve (C , {Σi }n ) i=1 S constructed above is an equivalence of categories. Moreover, this equivalence of categories is compatible with base change S ′ → S . We prove this theorem below. Before giving the proof, however, let us record the main consequences of this result that we will use. Fix integers g and n, and let Sg,n denote the ﬁbered category over Z which to any scheme S associates the groupoid of all (not necessarily stable) n– pointed genus g nodal curves C/S . The stack Sg,n is algebraic, and as explained in section [29, §5] the substack S0 ⊂ Sg,n classifying smooth g,n curves deﬁnes a log structure MSg,n on Sg,n . The same argument used in [29, §5] yields the following theorem: Theorem A.6. Let Mtw denote the ﬁbered category over Z which to any g,n scheme T associates the groupoid of n–marked genus g twisted curves (C , {Σi }) over T . Then Mtw is a smooth Artin stack, and the natural map g,n (A.6.1) π : Mtw −→ Sg,n g,n ♠ sending (C , {Σi }) to its coarse moduli space with the marked points induced by the Σi is representable by tame stacks. Moreover, there is a natural locally free log structure MMtw on Mtw . g,n g,n Remark A.7. Consider a ﬁeld k and an object (C , {Σi }) ∈ Mtw (k). Let g,n (C, {σi }) be the coarse moduli space, and let R be a versal deformation space for the object (C, {σi }) ∈ Sg,n (k). Let q1 , . . . , qm ∈ C be the nodes and let ri be the order of the stabilizer group of a point of C lying above qi . As in [11, 1.5], there is a smooth divisor Di ⊂ Spec(R) classifying deformations where qi remains a node. In other words, if ti ∈ R is an element deﬁning Di then in an ´tale neighborhood of qi the versal deformation C → Spec(R) of e (C, {σi }) is isomorphic to Spec(R[x, y ]/(xy − ti )). 44 ABRAMOVICH, OLSSON, AND VISTOLI It follows from the argument in [29, §5] that the ﬁber product Mtw ×Sg,n Spec(R) g,n is isomorphic to the stacktheoretic quotient of
r rm Spec(R[z1 , . . . , zm ]/(z11 − t1 , . . . , zm − tm )) by the action of µr1 × · · · µrm for which (ζ1 , . . . , ζr ) ∈ µr1 × · · · µrm sends zi to ζi zi . In particular Mtw is ﬂat over Sg,n , and hence also ﬂat over Z. g,n We also get a generalization of [29, 1.11]: Corollary A.8. For any integer N > 0, let Mtw,≤N denote the substack of g,n Mtw classifying n–pointed genus g twisted curves such that the order of the g,n stabilizer group at every point is less than or equal to N . Then Mtw,≤N is g,n an open substack of Mtw and the map Mtw,≤N → Sg,n is of ﬁnite type and g,n g,n quasiﬁnite. Remark A.9. Let R be a discrete valuation ring with uniformizer π and separably closed residue ﬁeld, and let C/R be a nodal curve. Let p1 , . . . , pr be the nodes of C in the closed ﬁber. Then we have a log smooth morphism (C, MC ) → (Spec(R), MR ), where the log structure MR → R admits a chart Nr → R such that the image of the ith standard generator is equal to π li , where li ∈ N ∪ {∞} is an element such that in an ´tale neighborhood of pi the curve Ci is isomorphic e to Spec(R[x, y ]/(xy − π li ), where by convention if li = ∞ we set π li = 0. Now assume some li is ﬁnite. Then we obtain a twisted curve by taking the stack corresponding to the morphism of log structures MR → M′ , R where M′ is the log structure associated to the map Nr → R sending ej to R π lj for j = i, and ei to π . Proof of A.5. The proof of A.5 follows the same outline as the proof of [29, 1.9]. We review the argument here indicating the necessary changes for this more general setting. Deﬁnition A.10 ([29, 3.3]). A log smooth morphism of ﬁne log schemes f : (X, MX ) → (S, MS ) is essentially semistable if for each geometric point x → X the monoids (f −1 M S )x and M X,x are free monoids, and if ¯ ¯ ¯ for suitable isomorphisms (f −1 M S )x ≃ Nr and M X,x ≃ Nr+s the map ¯ ¯ (f −1 M S )x → M X,x ¯ ¯ is of the form (A.10.1) ei → ei if i = r er + er+1 + · · · + er+s if i = r, where ei denotes the ith standard generator of Nr . TWISTED YET TAME 45 With notation as in A.10, if U → X is a smooth surjection of schemes and MU denotes the pullback of MX , then it follows immediately from the deﬁnition that (X, MX ) → (S, MS ) is essentially semistable if and only if the morphism (U, MU ) → (S, MS ) is essentially semistable (the property of being semistable is local in the smooth topology on X ). It follows that the notion of a morphism being essentially semistable extends to Artin stacks: If (X, MX) is an Artin stack with a ﬁne log structure, and f : (X, MX) → (S, MS ) is a morphism to a ﬁne log scheme, then f is essentially semistable if for some smooth surjection U → X with U a scheme the induced morphism of log schemes (A.10.2) (U, MXU ) → (S, MS ) is essentially semistable. As usual if for some smooth surjection U → X the morphism A.10.2 is essentially semistable then for any smooth surjection V → X the induced morphism (V, MXV ) → (S, MS ) is essentially semistable. As explained in [29, §3], if (X, MX ) → (S, MS ) is an essentially semistable morphism of log schemes, then for any geometric point s → S there is a ¯ canonical induced map sXs : {singular points of Xs } → Irr(MS,s ), ¯ ¯ ¯ where Irr(MS,s ) denotes the set of irreducible elements in the monoid MS,s . ¯ ¯ If Y is an Artin stack over a ﬁeld k, let π0 (Ysing ) denote the set of connected components of the complement of the maximal open substack U ⊂ Y which is smooth over k. Note that if Y′ → Y is a smooth morphism of kstacks, then there is an induced map π0 (Y′ ) → π0 (Ysing ). sing If Y′ → Y is also surjective then this map is also surjective. Deﬁnition A.11. Let f : (X, MX) → (S, MS ) be an essentially semistable morphism from a log Artin stack to a ﬁne log scheme, and let s → S be a ¯ geometric point. We say that f is special at s if for any smooth surjection ¯ U → X with U a scheme the map sUs : {singular points of Us } → Irr(MS,s ) ¯ ¯ ¯ factors through the composite (which is surjective) {singular points of Us } − − → π0 (Us,sing ) − − → π0 (Xs,sing ) −− −− ¯ ¯ ¯ to give an isomorphism π0 (Xs,sing ) → Irr(MS,s ). ¯ ¯ Theorem A.12 (Generalization of [29, 3.6]). Let (f : C → S, {Σi }) be an nmarked twisted curve. Then there exist log structures MC and M′ on C S and S respectively, and a special morphism (f, f b ) : (C , MC ) −→ (S, M′ ). S 46 ABRAMOVICH, OLSSON, AND VISTOLI Moreover, the datum (MC , M′ , f b ) is unique up to unique isomorphism. S Proof. The proof will be in several steps (ending in paragraph following A.17). The uniqueness statement follows as in [29, 3.6] from the argument proving the uniqueness in [31, 2.7]. Given the uniqueness, to prove existence we may work ´tale locally on S , e and by a limit argument as in the proof of [29, 3.6] may even assume that S is the spectrum of a strictly henselian local ring. Let s ∈ S be the closed ¯ point. Let p1 , . . . , pn ∈ C be the nodes of the closed ﬁber, and choose for each i = 1, . . . , n an aﬃne open set Ui ⊂ C containing pi and no other nodes. Let Ui ⊂ C denote the inverse image of Ui . Lemma A.13. For any quasicoherent sheaf F on Ui , we have H j (Ui , F ) = 0 for j > 0. Proof. This follows from the same argument proving 2.5. Let ti ∈ OS be an element such that the ﬁber product C ×C Spec(OC,pi ) ¯ is isomorphic to [Spec(OS [z, w]/(zw − ti ))/µn ] as in (2.1 (v)). Let be the log structure on S associated to the morphism N → OS sending 1 to ti Mi S Deﬁnition A.14. Let f : X → S be a morphism of schemes. A ti semistable log structure on X is a pair (MX , f b ), where MX is a locally free log structure on X and f b : f ∗ Mi → X is a morphism of log strucS tures, such that the following hold: (i) The morphism of log schemes (f, f b ) : (X, MX ) → (S, Mi ) S is log smooth; (ii) For every geometric point x → X the induced map of free monoids ¯ N → MS,f (¯) → MX,x ¯ x is the diagonal map. Remark A.15. By [29, 3.4], if X → S admits a ti semistable log structure, then ´tale locally on X there exists a smooth morphism e X → Spec(OS [z, w]/(zw − ti )). To prove A.12, it suﬃces by the same argument used in [29, proof of 3.6] to show that there exists a ti semistable log structure on each Ui .
i ♠ TWISTED YET TAME 47 Lemma A.16. Let X → Ui be a smooth morphism with X a scheme. Then ´tale locally on X there exists a ti semistable log structure. e Proof. It suﬃces to prove the existence in some ´tale neighborhood of a e geometric point x → X mapping to the node pi of C . Making an ´tale base ¯ e change on C , it therefore suﬃces to show that if g : X → [Spec(OS [z, w]/(zw − ti )/µn ] is a smooth morphism then there exists a ti semistable log structure on X . For this note that there is a log structure M on [Spec(OS [z, w]/(zw − ti )/µn ] and a morphism of log stacks (A.16.1) ([Spec(OS [z, w]/(zw − ti )/µn ], M) → (S, MS )
β induced by the commutative diagram −− N2 − − → OS [z, w]/(zw − ti ) ∆ i −− N −−→ 1→t OS where β sends (1, 0) to z and (0, 1) to w. By [32, 5.23], the morphism A.16.1 is log ´tale. It follows that the pullback g∗ M on X is a ti semistable log e structure on X . ♠ e Let SSti denote the presheaf on the lisse´tale site LisEt(Ui ) of Ui which to any smooth morphism X → Ui associates the set of isomorphism classes of ti semistable log structures on X . As in the proof of [31, 3.18] a ti semistable log structure admits no nontrivial automorphisms, and hence SSti is in fact a sheaf. In fact, SSti is a torsor under a certain sheaf of abelian groups which we now describe. For any smooth morphism X → Ui , there exists by A.16 and A.15 ´tale e locally on X a smooth morphism ρ : X → Spec(OS [z, w]/(zw − ti )). As explained in [31, 3.12] the ideal J := (z, w)OX ⊂ OX is independent of the choice of the smooth morphism ρ. It follows that these locally deﬁned sheaves of ideals descend to a sheaf of ideals J ⊂ OUi . Let D ⊂ Ui be the closed substack deﬁned by this sheaf of ideals. The local description (2.1 (v)) of the stack Ui implies that there is an isomorphism D ≃ Bµn ×Spec(Z) Spec(OS /(ti )).
U Let Kti ⊂ OS be the kernel of multiplication by ti on OS , and let Kti i denote the kernel of multiplication by ti on OUi . Note that since Ui is ﬂat U over S the sheaf Kti i is equal to the pullback of Kti . Let Z ⊂ U be the U closed substack deﬁned by Kti i · J . Deﬁne ∗ ∗ G := Ker(OUi → OZ ), 48 ABRAMOVICH, OLSSON, AND VISTOLI ∗ and let G2 ⊂ OUi denote the subsheaf of units u such that uti = ti . There is a natural inclusion G ⊂ G2 , and as explained in the proof of [31, 3.18] the sheaf SSti is naturally a torsor under G2 /G. To prove the existence of a ti semistable log structure on Ui we therefore must show that the class of this torsor o ∈ H 1 (Ui , G2 /G) is zero. ∗ Lemma A.17. The map H 1 (Ui , G2 /G) → H 1 (D , OD ) induced by the composite ∗ ∗ G2 ⊂ OUi → OD is injective. Proof. This follows from the same argument proving [29, 3.7] (and using A.13). ♠
∗ The image of the class o in H 1 (D , OD ) corresponds to an invertible sheaf L on D . As in [31, proof of 3.16] this invertible sheaf L can be described as follows. Consider the inclusion Bµn ֒→ [Spec(OS [z, w]/(zw − ti ))/µn ] ≃ Ui ×Ui Spec(OC,pi ). ¯ Then L corresponds to the representation of µN with basis z w . By the ˙ assumptions on the µn action in (2.1 (v)) it follows that L is trivial, and hence o = 0. ♠ As in [29, 3.10], the log structure MC is not the “right” log structure on C as it does not take into account the markings. Exactly as in loc. cit., for each i = 1, . . . , n the ideal sheaf Ii ⊂ OC deﬁning Σi deﬁnes a log structure Ni on C . Set ∗ MC := MC ⊕OC (⊕n ,O∗ Ni ). i=1
C The map f ∗ M′ S → MC then induces a log smooth morphism (C , MC ) → (S, M′ ). S If g : (C, σ1 , . . . , σn ) → S is the coarse moduli space of C with its n sections deﬁned by the pi , then the above construction applied to (C, σ1 , . . . , σn ) yields log structures MC and MS on C and S respectively and a special morphism (C, MC ) → (S, MS ). Proposition A.18 (Generalization of [29, 4.7]). Let π : C → C be the projection. There exists canonical morphisms of log structures π b : π ∗ MC → MC and ℓ : MS ֒→ M′ such that the diagram of log stacks S (C , MC ) − − → (C, MC ) −− g f
(id,ℓ) (π,π b ) (S, M′ ) − − → (S, MS ) −− S TWISTED YET TAME 49 commutes. Moreover, the map ℓ : MS ֒→ M′ is a simple extension. S Proof. This follows from the same argument proving [29, 4.7]. The key point is the local description (2.1 (iv) and (v)) of a twisted curve which enables one to rewrite [29, 4.6] verbatim in the present situation. ♠ Finally note that for any geometric point s → S the gerbe Σi,s is neces¯ ¯ sarily trivial and hence isomorphic to Bµai (¯) for some integer ai (¯). One s s veriﬁes immediately that the ai are positive integer valued locally constant functions on S . The association (C , {Σi }) → (C/S, {σi , ai }, ℓ : MS ֒→ M′ ) S therefore deﬁnes a functor (A.18.1) (twisted npointed curves) → (npointed log twisted curves) As in [29, 4.8] one sees that A.5.1 and A.18.1 are inverse functors thereby proving A.5. ♠ Appendix B. Remarks on Extgroups and base change By Martin Olsson In this appendix we gather together some fairly standard results about Extgroups and base change, which will be used in the following appendix C. The results B.2 and B.10 are wellknown to experts, but we include them here due to lack of a suitable reference. B.1. Boundedness of cohomology. Theorem B.2. Let f : X → Y be a separated, representable, ﬁnite type morphism between noetherian algebraic stacks. Then there exists an integer n0 such that for any quasicoherent sheaf F on X and i ≥ n0 we have Ri f∗ F = 0. Proof. The assertion is fppf local on Y , so we may assume that Y is a noetherian aﬃne scheme, and that X is an algebraic space. In this case we must show that there exists an integer n0 such that for any quasicoherent sheaf F on X and i ≥ n0 we have H i (X, F ) = 0. Lemma B.3. Let Z0 ֒→ Z be a closed immersion of noetherian algebraic spaces over Y deﬁned by an nilpotent ideal J ⊂ OZ . Assume that B.2 holds for Z0 → Y and let nZ0 be an integer such that for any quasicoherent sheaf F0 on Z0 we have H i (Z0 , F0 ) = 0 for i ≥ nZ0 . Then for any quasicoherent sheaf F on Z we have H i (Z, F ) = 0 for i ≥ nZ0 . Proof. Let F be a quasicoherent sheaf on Z . Let n be an integer such that Jn = 0, and set Fk = Jk F so that we have an increasing sequence of quasicoherent sheaves 0 = Fn ⊂ Fn−1 ⊂ · · · F0 = F 50 ABRAMOVICH, OLSSON, AND VISTOLI with successive quotients Fk /Fk+1 supported on Z0 . The result then follows by descending induction on k and consideration of the short exact sequences 0 → Fk+1 → Fk → Fk /Fk+1 → 0 which gives rise to exact sequences H i (Z, Fk+1 ) → H i (Z, Fk ) → H i (Z0 , Fk /Fk+1 ), where the last term is zero by assumption. ♠ We now prove B.2 by noetherian induction. By [23, II.6.7] there exists a dense open subspace j : U ⊂ X with U an aﬃne scheme. Let Z ⊂ X be the complement (with the reduced structure). Let nZ be an integer such that for any quasicoherent sheaf FZ on Z we have H i (Z, FZ ) = 0 for i ≥ nZ . Lemma B.4. Let G be a quasicoherent sheaf on X whose restriction to U is the zero sheaf. Then H i (X, G ) = 0 for i ≥ nZ . Proof. By [23, III.1.1] the sheaf G is equal to the limit G = lim Gi over its − → coherent subsheaves Gi . Furthermore by [23, II.4.17] we have H i (X, G ) = lim H i (X, Gi ). − → It follows that it suﬃces to consider the case when G is coherent. In this case there exists a nilpotent thickening Z ⊂ Z ′ ⊂ X of Z in X such that the schemetheoretic support of G is contained in Z ′ . The result therefore follows from B.3. ♠ Since U is aﬃne and X is separated, the morphism j : U → X is an aﬃne morphism. It follows that the sheaf j∗ j ∗ F is a quasicoherent sheaf on X and that H i (X, j∗ j ∗ F ) = H i (U, j ∗ F ). Since U is aﬃne these groups are zero for i > 0. Set nX := nZ + 1. Let K (resp. I , Q) be the kernel (resp. image, cokernel) of the adjunction map F → j∗ j ∗ F . By B.4 we have H i (X, K ) and H i (X, Q) equal to 0 for i ≥ nZ . Consideration of the long exact sequences associated to the short exact sequences 0→K→F →I→0 and 0 → I → j∗ j ∗ F → Q → 0 then shows that H i (X, F ) = 0 for i ≥ nX . ♠ TWISTED YET TAME 51 B.5. Base change for Ext. Let f : X → S be a ﬁnite type morphism between noetherian algebraic stacks, and assume that S is an integral scheme. − Fix L· ∈ Dcoh (X ) (the derived category of bounded above complexes of OX modules with coherent cohomology sheaves) and a coherent sheaf J ∈ Coh(X ). Lemma B.6. For every integer n, there exists a dense open subset (which depends on n) U ⊂ S such that for any cartesian diagram −− X ′ −−→ X f ′ −− S ′ − − → S,
g h f where g factors through U , the natural map h∗ E xtp (L· , J ) → E xtp (Lh∗ L· , Lh∗ J ) is an isomorphism for p ≤ n. Proof. The assertion is local in the ﬂat topology on X , S , and S ′ . We may therefore assume that X = Spec(R) and S = Spec(A) are aﬃne schemes, and that L· can be represented by a bounded above complex of projective Rmodules of ﬁnite type, which we again denote by L· . Let J denote the Rmodule corresponding to the sheaf J , and let F · denote the bounded below complex of ﬁnite type Rmodules F · := Hom· (L· , J ). After shrinking on S we may assume that J is ﬂat over A, in which case each F j is ﬂat over A. We need to show that after possibly replacing S by a dense aﬃne open subset, the natural map H p (F · ) ⊗A A′ → H p (F · ⊗A A′ ) is an isomorphism for all ring homomorphisms A → A′ and all p ≤ n. This is a standard argument and we leave it to the reader (see for example [14, IV.9.4.3] where a similar argument is made). ♠ We would like to use this result to obtain base change properties of global Extgroups. In order for this to be possible, however, we need certain ﬁniteness properties of cohomology. Deﬁnition B.7. Let f : X → Y be a morphism of noetherian algebraic stacks over a scheme S . b (i) f is cohomologically bounded if for every object F ∈ Dqcoh (X ) there exists an integer n such that Ri f∗ F = 0 for i ≥ n. (ii) f is universally cohomologically bounded if for every morphism of noetherian algebraic stacks Y ′ → Y the morphism f ′ : X ′ := X ×Y Y ′ → Y ′ is cohomologically bounded. 52 ABRAMOVICH, OLSSON, AND VISTOLI Remark B.8. By a standard devissage, f is cohomologically bounded if and only if for every quasicoherent sheaf F on X there exists an integer n such that Ri f∗ F = 0 for i ≥ n. Remark B.9. Note that any separated representable morphism f : X → Y is universally cohomologically bounded by B.2. Theorem B.10. Let f : X → S be a ﬂat ﬁnite type proper morphism of noetherian algebraic stacks with S = Spec(A) an integral aﬃne scheme, − and assume that f is cohomologically bounded. Let L· ∈ Dcoh (X ) and J ∈ Coh(X ). Then for every integer n, there exists a dense open subset (which depends on n) U ⊂ S such that for any cartesian diagram X′ ′ −−→ X −− f
g h f S ′ = Spec(A′ ) − − → S, −− where g factors through U , the natural map (B.10.1) Extp (L· , J ) ⊗A A′ → Extp (Lh∗ L· , Lh∗ J ) is an isomorphism for p ≤ n. Proof. This follows from consideration of the localtoglobal spectral sequences for Ext from B.2, and standard base change properties for cohomology of coherent sheaves (see for example [30, 5.12]). First after shrinking on S we may assume that J is ﬂat over S and that for any morphism g : Spec(A′ ) → Spec(A) the pullback map h∗ E xtp (L· , J ) → E xtp (Lh∗ L· , Lh∗ J ) is an isomorphism for all p ≤ n. By standard base change properties of cohomology of coherent sheaves as in [30, 5.12] (note that this is where the cohomological boundedness is used), it follows that after shrinking some pq more on S we may assume that each of the terms Er for p + q ≤ n in the spectral sequence
pq E2 = H p (X , E xtq (L· , J )) =⇒ Extp+q (L· , J ) are ﬂat over S , and that their formation commute with arbitrary base change Spec(A′ ) → Spec(A). That B.10.1 is an isomorphism then follows from consideration of the morphism of spectral sequences
pq E2 = H p (X , E xtq (L· , J )) =⇒ Extp+q (L· , J ) pq E2 = H p (X ′ , E xtq (Lh∗ L· , Lh∗ J )) =⇒ Extp+q (Lh∗ L· , Lh∗ J ). ♠ TWISTED YET TAME 53 Appendix C. Another boundedness theorem for Homstacks By Martin Olsson C.1. Statement of Theorems. C.2. Let B be a scheme, and X and Y be Artin stacks of ﬁnite presentation over B with ﬁnite diagonals. Let X and Y denote the coarse moduli spaces of X and Y . Assume that X is ﬂat and proper over B , and that fppf locally on B there exists a ﬁnite and ﬁnitely presented ﬂat surjection Z → X with Z an algebraic space. By [30, 1.1], we then have Artin stacks Hom(X , Y ) and Hom(X , Y ) locally of ﬁnite presentation over B with separated and quasi–compact diagonals. Theorem C.3. Assume that Y is a tame stack. Then the natural map Hom(X , Y ) → Hom(X , Y ) is of ﬁnite type. There is also a variant of C.3, where one does not assume that X admits a ﬁnite ﬂat cover by a scheme but instead that X is tame (this is in fact the theorem we will apply to twisted stable maps): Theorem C.4. Let B be a scheme locally of ﬁnite type over an excellent Dedekind ring, and let X and Y be tame Artin stacks of ﬁnite presentation over B with ﬁnite diagonals. Let X (resp. Y ) denote the moduli space of X (resp. Y ). Assume that X is ﬂat and proper over B . Then Hom(X , Y ) is an algebraic stack locally of ﬁnite presentation over B with quasicompact and separated diagonal, and the map Hom(X , Y ) → Hom(X , Y ) = Hom(X, Y ) is of ﬁnite type. Remark C.5. Note that by [4, 3.3], for any morphism B ′ → B the base change XB ′ → XB ′ is the moduli space of XB ′ . Remark C.6. The assumption that X /B is ﬂat implies that X/B is also ﬂat by [4, 3.3 (b)]. It follows that Hom(X, Y ) is an algebraic space locally of ﬁnite presentation over B . In the context of either C.3 or C.4, one can also consider the substack Homrep (X , Y ) ⊂ Hom(X , Y ) classifying representable morphisms X → Y . As in [30, 1.6] this substack is an open substack, and therefore the following corollary follows from C.3 and C.4: Corollary C.7. The natural map Homrep (X , Y ) → Hom(X , Y ) is of ﬁnite type. 54 ABRAMOVICH, OLSSON, AND VISTOLI The rest of this appendix is devoted to the proofs of C.3 (paragraphs C.17C.30) and C.4 (paragraphs C.31C.36). C.8. We begin with the proof of C.3. As in [35, 6.2], Theorem C.3 is equivalent to the following statement. For any morphism f : X → Y the stack Sec(X ×Y Y /X ) which to any B –scheme T associates the groupoid of sections s : X → X ×Y Y is of ﬁnite type. It is this latter statement that we will prove. Let G denote X ×f,Y Y . The proof that Sec(G /X ) is of ﬁnite type over B follows the same outline as in [35]. First (in paragraphs C.17–C.22) we reduce the proof of C.3 to the special case when X = X and G is a µn –gerbe for some integer n. We then prove the theorem in this special case (paragraphs C.23–C.36) using some relatively straightforward generalizations to “twisted sheaves” of theorems about the Picard functor. The main new wrinkle is to use the base change properties of Ext explained in appendix B. C.9. Some results about modiﬁcations. We use the results of appendix B to generalize three basic results about modiﬁcations in [35, §4] to tame stacks. Passage to the maximal reduced substack. C.10. Let S be a noetherian scheme, and let G → X be a ﬁnite type morphism between Artin stacks of ﬁnite type over S with ﬁnite diagonals. Assume further that X is ﬂat and proper over S and that the structure morphism X → S is universally cohomologically bounded in the sense of B.7. Assume further that fppf locally on S there exists a ﬁnite ﬂat cover of X by an algebraic space. Proposition C.11. Let X0 ֒→ X be a closed immersion deﬁned by a nilpotent ideal J ⊂ OX , and assume X0 is ﬂat over S . Let G0 denote the base change G ×X X0 . Then the natural map Sec(G /X ) → Sec(G0 /X0 ) is of ﬁnite type. Proof. This is essentially the same as in [30, 5.11]. By noetherian induction, it suﬃces to show that the morphism is of ﬁnite type over a dense open subset of S . We may further assume that S is reduced (since if U and V are S schemes locally of ﬁnite type and g : U → V is a morphism, then g is of ﬁnite type if and only if the base change of g to Sred is of ﬁnite type), and using the same argument given in [30, paragraph following proof of 5.11] that J 2 = 0. It suﬃces to show that if T → Sec(G0 /X0 ) is a morphism corresponding to a section s : X0 → G0 , then the ﬁber product P := Sec(G /X ) ×Sec(G0 /X0 ) T TWISTED YET TAME 55 is of ﬁnite type over T . Furthermore, we may assume that T is an integral noetherian aﬃne scheme. The stack P associates to any w : W → T the groupoid of liftings s : XW → G over X of the composite ˜ X0,W → G0 → G , where the ﬁrst map is the one induced by s. To prove that P is quasi–compact it suﬃces by Noetherian induction to exhibit a dense open set U ⊂ T such that PU is quasi–compact. Let LG0 /X0 be the cotangent complex of G0 /X0 . By B.10 (this is where the assumption of cohomological boundedness is used), after replacing T by a dense open subscheme we may assume that the groups Exti (s∗ LG0 /X0 , J ), i = −1, 0, 1 are projective modules on T of ﬁnite type, and that the formation of these modules commutes with arbitrary base change on T . By [34, 1.5] there is a canonical obstruction o ∈ Ext1 (s∗ LG0 /X0 , J ) whose vanishing is necessary and suﬃcient for the existence of a lifting s ˜ as above, and the formation of this obstruction is functorial in T . After replacing T by the closed subscheme deﬁned by the condition that o vanishes, we may assume that o = 0. In this case the set of isomorphism classes of liftings s form a torsor under ˜ Ext0 (s∗ LG0 /X0 , J ) and the group of inﬁnitessimal automorphisms of s is canonically isomorphic ˜ to Ext−1 (s∗ LG0 /X0 , J ). It follows that P is an Ext−1 (s∗ LG0 /X0 , J )–gerbe over Ext0 (s∗ LG0 /X0 , J ), and in particular is quasi–compact. The case of a ﬁnite morphism. C.12. Let S be a noetherian scheme, and X /S a proper ﬂat Artin stack with ﬁnite diagonal. Let G → X be a ﬁnite morphism. Assume that fppf locally on S , the stack X admits a ﬁnite ﬂat surjection Z → X with Z an algebraic space. Proposition C.13. If X → S is universally cohomologically bounded, then the stack Sec(G /X ) is of ﬁnite type and separated over S . Proof. To verify that Sec(G /X ) is of ﬁnite type over S , it suﬃces to show that its pullback to Sred is of ﬁnite type. We may therefore assume that S is reduced. Furthermore, using the fact that X → S is universally cohomologically bounded, we may by C.11 (applied with X0 = Xred ) assume that X is ♠ 56 ABRAMOVICH, OLSSON, AND VISTOLI also reduced. ) Furthermore, by noetherian induction it suﬃces to exhibit a dense open subset of S over which Sec(G /X ) is of ﬁnite type. By Chow’s lemma for Artin stacks [33, 1.1], there exists a proper surjection h : X ′ → X with X ′ a projective S scheme. After shrinking on S we may also assume that X ′ is ﬂat over S (since S is reduced). Since X is reduced the map OX → h∗ OX ′ is injective. After shrinking on S , we may assume that the formation of h∗ OX ′ commutes with arbitrary base change S ′ → S , and that for any such base change the map OXS ′ → h∗ OX ′ ′ is also S injective. Let G′ → X ′ denote the pullback G ×X X ′ . By [30, 5.10] the stack (in fact an algebraic space) Sec(G′ /X ′ ) is of ﬁnite type over S . It therefore suﬃces to show that the pullback map (C.13.1) Sec(G /X ) → Sec(G′ /X ′ ) is of ﬁnite type. Let A be the coherent sheaf of algebras deﬁned by G = SpecX (A), and let B denote the sheaf of OX algebras h∗ OX ′ . For a morphism T → S , let AT (resp. BT ) denote the pullback of A (resp. B ) to XT := X ×S T . Then Sec(G′ /X ′ ) is equal to the functor which to any scheme T → S associates the set of morphisms of OXT algebras ϕ : AT → BT over XT . The stack Sec(G /X ) is the subfunctor of morphisms ϕ which factor through OXT ⊂ BT . Fix a morphism ϕ : AT → BT over a noetherian scheme T deﬁning a T valued point of Sec(G′ /X ′ ), and set P := Sec(G /X ) ×Sec(G′ /X ′ ),ϕ T. To prove that P is of ﬁnite type, we may again assume that T is reduced, and by noetherian induction it suﬃces to show that P → T is of ﬁnite type over a dense open subset of T . Let M denote the cokernel of the map OXT ֒→ BT , and let ϕ : AT → M be the map induced by ϕ. Let Q be the ¯ cokernel of ϕ, and let K be the kernel of the map M → Q so that there is ¯ an exact sequence (C.13.2) 0 → K → M → Q → 0. After shrinking on T we may assume that Q is ﬂat over T in which case C.13.2 remains exact after arbitrary base change T ′ → T . In this case, let Z ⊂ XT be the support of the coherent sheaf K , and let W ⊂ T be the image of Z (which is closed since XT → T is proper). Then P is represented by the complement of W in T . To verify that Sec(G /X ) is separated, note that we already know that the diagonal is quasicompact and separated. Therefore it suﬃces to verify the valuative criterion for properness. This amounts to the following. Assume that S is the spectrum of a valuation ring and let η ∈ S be the generic point. Assume given two sections s1 , s2 : X → G whose restrictions to η are equal. Then we need to show that s1 = s2 . For this note that since G is ﬁnite over X , we have G = SpecX (A) for some coherent sheaf of OX algebras A on X , and the sections s1 and s2 are speciﬁed by two morphisms of OX algebras TWISTED YET TAME 57 ρ1 , ρ2 : A → OX . Let j : Xη ֒→ X be the inclusion of the generic ﬁber. Since X is ﬂat over S , the natural map OX → j∗ OXη is an inclusion. Therefore it suﬃces to show that the composite maps A
ρi // OX // j∗ OXη are equal. Equivalently that the maps on the generic ﬁber ρi,η : Aη → OXη are equal, which holds by assumption. Behavior with respect to proper modiﬁcations of X . C.14. Let S be a noetherian scheme, and let X and Y be Artin stacks of ﬁnite type over S with ﬁnite diagonals. Assume that the following conditions hold: (i) The formation of the coarse spaces πX : X → X and πY : Y → Y commutes with arbitrary base change S ′ → S . (ii) X is proper and ﬂat over S , and fppf locally on S there exists a ﬁnite ﬂat surjection Z → X with Z an algebraic space. (iii) The coarse space X of X is ﬂat over S (note that X is automatically proper over S since X is proper over S ). (iv) The morphism X → S is universally cohomologically bounded. Proposition C.15. Let m : X ′ → X be a proper representable surjection with X ′ a proper and ﬂat algebraic stack over S with ﬁnite diagonal. Let f : X → Y be a morphism, and let G (resp. G ′ ) denote the pullback along f (resp. f ◦ m) of Y . Assume that Sec(G /X ) and Sec(G ′ /X ′ ) are algebraic stacks locally of ﬁnite type over S with quasicompact diagonals. Then the pullback map (C.15.1) is of ﬁnite type. Remark C.16. Note that since m is representable, the stack X ′ also admits a ﬁnite ﬂat surjection from an algebraic space. Proof. This follows from the same argument proving [35, 4.13]. The cohomological boundedness assumption is necessary in order to apply C.11. ♠ C.17. D´vissage. We now begin the proof of the statement that Sec(G /X ) e in C.8 is of ﬁnite type over B . In the rest of the proof that Sec(G /X ) is of ﬁnite type over B , we work under the assumptions of C.14. By a standard limit argument as discussed for example in [30, §2] we can without loss of generality assume that B is of ﬁnite type over Z. C.18. The assertion that Sec(G /X ) is of ﬁnite type is fppf local on B , and therefore we may assume that there exists a ﬁnite ﬂat surjection Z → X with Z an algebraic space. Let Z (i) denote the i–fold ﬁber product of Z with itself over X . Then by the description of Sec(G /X ) in terms of the Sec(G /X ) → Sec(G ′ /X ′ ) ♠ 58 ABRAMOVICH, OLSSON, AND VISTOLI Sec(G ×X Z (i) /Z (i) ) given in [30, 3.3] it suﬃces to show that for i = 1, 2, 3 the stacks Sec(G ×X Z (i) /Z (i) ) are of ﬁnite type. We may therefore assume that X = X . Furthermore, we can without loss of generality assume that B is integral. By noetherian induction, it suﬃces to exhibit a dominant morphism B ′ → B such that the restriction of Sec(G /X ) to B ′ is of ﬁnite type. C.19. By the same argument used in [35, 6.4] we then reduce the proof of C.3 to the case when X is integral and G → X has a section s : X → G . Let Aut(s) denote the group scheme of automorphisms of s (a scheme over X ). Since X is reduced there exists a dense open subset U ⊂ X such that the restriction of Aut(s) to U is ﬂat over U . Let H ⊂ Aut(s) denote the scheme– theoretic closure of Aut(s)U . By [17, Premi´re partie 5.2.2] there exists a e blowup X ′ → X with center a proper closed subspace of X such that the strict transform of H in Aut(s) is ﬂat over X ′ . After further shrinking on S we may assume that X ′ is ﬂat over S . Since the map Sec(G /X ) → Sec(G ×X X ′ /X ′ ) is of ﬁnite type by C.15, we may therefore replace X by X ′ and hence can assume that H is a ﬁnite ﬂat subgroup scheme of Aut(s). Since all the geometric ﬁbers of H are linearly reductive being closed subgroup schemes of linearly reductive groups (see [4, 2.7]), the X group scheme H is in fact linearly reductive. If X is the normalization of X the map Sec(G /X ) → Sec(G ×X X /X ) is also of ﬁnite type by C.15. This enables us to reduce to the following situation: X is normal, s : X → G is a section, and H ⊂ Aut(s) is a ﬁnite closed subgroup scheme which is ﬂat and linearly reductive over X and over a dense open subset of X the group scheme H is equal to Aut(s). Lemma C.20. The morphism π : BH → G induced by s identiﬁes BH with the normalization of Gred . Proof. Etale locally on X we can write G = [W/GLn ] for some n, where W is an X –scheme. Indeed by [4, 3.2] we can ´tale locally on X write G = e [Z/G] with G a linearly reductive group scheme. Choosing any embedding G ֒→ GLn we take W to be the quotient of Z × GLn by the diagonal action of G. Let P → X be the pullback X ×s,G W . Then P is a GLn –torsor over X with a GLn –equivariant morphism f : P → W . Since P is smooth over X the space P is in particular normal, and since s is proper and quasi–ﬁnite the morphism f is ﬁnite. After replacing X by an ´tale cover we may assume e given a section p ∈ P . Then Aut(s) is the closed subgroup scheme of GLn,X TWISTED YET TAME 59 W × W ← − − X, −− where ∆ : W → W × W is the diagonal and ρ is the map sending (v, g) to (v, g(v )). In particular, we obtain an embedding H ⊂ GLn,X . Let P denote the quotient P/H . The space P is normal. By the construction the map f ¯ induces a morphism f : P → W ∗ , where W ∗ denotes the normalization of ¯ Wred . This map f is ﬁnite, surjective, and birational and hence by Zariski’s Main Theorem an isomorphism. On the other hand, we have P ≃ W ×G BH , and hence if G ∗ denotes the normalization of Gred we ﬁnd that the base change of BH → G ∗ to W ×G G ∗ is an isomorphism. It follows that BH → G ∗ is an isomorphism. ♠ C.21. Let G ∗ denote the normalization of G , so that G ∗ is a gerbe over X . As in [35, 3.14], after shrinking on B we may assume that for every ﬁeld ∗ valued point b ∈ B (k) the ﬁber Gb → Gb is the normalization of Gb,red and that Xb is normal. It then follows that the map Sec(G ∗ /X ) → Sec(G /X ) is surjective on ﬁeld valued points and hence surjective. We may therefore replace G by G ∗ and therefore may assume that G is a gerbe over X . Next we consider the case of a gerbe. If the generic point of B has characteristic 0, then after shrinking on B we can assume G is Deligne– Mumford in which case the result follows from [35, 1.1]. We may therefore assume that the generic point of B has characteristic p > 0, and hence after shrinking on B may assume that B is an Fp –scheme. In this case, for any B –scheme T and t ∈ G (T ), the automorphism group scheme Gt of t is canonically an extension 1 → ∆t → Gt → Ht → 1, where ∆t is locally diagonalizable and Ht is ´tale over T . Indeed, the group e scheme Gt is tame so we can take ∆t to be the subfunctor of Gt classifying elements of Gt killed by some power of p. This normal subgroup ∆t is functorial in the pair (T, t). Deﬁne H to be the stack (with respect to the fppf topology) associated to the prestack whose objects are the same as the objects of G but for which a morphism between two t, t′ ∈ G (T ) is deﬁned to be the set ∆t \Hom(t, t′ ) = ∆t \Hom(t, t′ )/∆t′ = Hom(t, t′ )/∆t′ . Then H is a Deligne–Mumford stack and there is a natural map G → H over X . By [35, 1.1] the stack Sec(H/X ) is of ﬁnite type. On the other hand, for any section s : X → H, the ﬁber product Sec(G /X ) ×Sec(H/X ),s B given by the ﬁber product of the diagram W × GLn ρ ∆◦f ◦p 60 ABRAMOVICH, OLSSON, AND VISTOLI is isomorphic to Sec(G ×H,s X/B ). Now the stack G ×H,s X is a gerbe over X whose stabilizer groups are all diagonalizable. This further reduces the proof to the case when the Ht ’s in the above discussion are all trivial. C.22. In this case, there exists a canonical locally diagonalizable group scheme ∆ on X such that G is bound by ∆. Indeed fppf locally on X there exists a section s : X → G giving rise to a group scheme ∆s . If s′ : X → G is a second section then locally s and s′ are isomorphic so we obtain an isomorphism ∆s ≃ ∆s′ . This isomorphism is independent of the choice of the isomorphism between s and s′ since ∆s and ∆s′ are abelian. It follows that the ∆s ’s descend to a group scheme ∆ on X . Since the Cartier dual of ∆ is a locally constant sheaf of ﬁnite abelian groups on Xet , there exists a ﬁnite ´tale covering X ′ → X such that the e ′ is a diagonalizable group scheme. Since pullback of ∆ to X Sec(G /X ) → Sec(G ×X X ′ /X ′ ) is of ﬁnite type by C.15, this reduces the proof to the case when G is a gerbe over X bound by a diagonalizable group scheme ∆. In other words, when G corresponds to a cohomology class in H 2 (X, ∆). Write ∆ = µn1 × · · · × µnr so that H 2 (X, ∆) = H 2 (X, µni ).
i Using the resulting decomposition of the cohomology class of G , we see that G is isomorphic to a product G1 ×X · · · ×X Gr , where Gi is a µni –gerbe over X . Then Sec(Gi /X ). Sec(G /X ) ≃
i This therefore ﬁnally reduces the proof to the case of a µn –gerbe over X which is the case treated in the following subsection. C.23. G –twisted line bundles. C.24. Fix an integer n. Assume that X = X and that G is a µn –gerbe over X . Assume furthermore that if f : X → B denotes the structural morphism then the map OB → f∗ OX is an isomorphism, and the same remains true after arbitrary base change B ′ → B . This implies that for any scheme T and object s ∈ G (T ) we have an isomorphism µn ≃ Aut(s) and these isomorphisms are functorial in the pair (T, s). If L is a line bundle on G then for any such pair (T, s) we obtain a line bundle s∗ L on T which also comes equipped with an action of µn = Aut(s). The line bundle L is called a G –twisted invertible sheaf on X if this action of µn coincides with the standard action induced by the embedding µn ֒→ Gm . Note that if χ denotes the standard character µn ֒→ Gm then for any line bundle L on G the action of µn = Aut(s) on s∗ L is given by χi for some integer i (locally constant on T ). It follows that to check that a line bundle L is G –twisted it suﬃces to verify that the two actions coincide for pairs (T, s) with T the spectrum of an algebraically closed ﬁeld. TWISTED YET TAME 61 If L is a G –twisted sheaf on X , then L⊗n descends canonically to an invertible sheaf on X since the stabilizer actions of µn are trivial. We usually write just L⊗n for the sheaf on X obtained from L⊗n . Proposition C.25. There is a natural equivalence of stacks between Sec(G /X ) and the stack which to any B –scheme T associates the groupoid of pairs (L, ι), where L is a G –twisted sheaf on X and ι : L⊗n ≃ OX is an isomorphism of invertible sheaves on X . Proof. If s : X → G is a section, then there is a canonical action of µn = Aut(s) on F := s∗ OX and hence F decomposes canonically as ⊕χ∈Z/(n) Fχ . Lemma C.26. Each Fχ is locally free of rank 1 on G and for any two χ, ǫ ∈ Z/(n) the natural map Fχ ⊗ Fǫ → Fχ+ǫ is an isomorphism. If χ1 : µn → Gm denotes the standard inclusion, then Fχ1 is a G –twisted sheaf on X . Proof. The choice of the section s identiﬁes G with B Aut(s) ≃ Bµn × X . The ﬁber product X ×s,Bµn ,s X is equal to X × Bµn from which we see that s∗ F is equal to OX ⊗Z Z[X ]/(X n − 1) with the natural action of µn . From this the lemma follows. ♠ In particular, by sending a section s : X → G to Fχ1 with the isomorphism ⊗ Fχ1n ≃ OG we obtain a functor (C.26.1) Sec(G /X ) → (stack of pairs (L, ι) as in C.25). Conversely, given a G –twisted sheaf L with an isomorphism ι : L⊗n ≃ OX n− we can form the cyclic algebra A = ⊕i=01 L⊗i with multiplication induced by ⊗i ⊗ L⊗j → L⊗i+j and the map ι. We can then consider the natural maps L SpecG (A) → G . If G = Bµn × X and X → G is the map deﬁned by the trivial torsor, then the restriction of (L, ι) to X is an invertible sheaf L on X with an isomorphism n− L⊗n ≃ OX and the restriction of A to X is just the cyclic algebra ⊕i=01 L⊗i ⊗i through the character u → ui . It follows that the with µn acting L projection SpecG (A) → X is an isomorphism and hence deﬁnes a point of Sec(G /X ). We leave to the reader that this deﬁnes an inverse to C.26.1. ♠ Let PicG X/B denote the stack over B which to any B –scheme T associates the groupoid of GT –twisted invertible sheaves on XT (where GT denotes G ×B T etc.). Proposition C.27. (i) The stack PicG X/B is an algebraic stack locally of ﬁnite presentation over B . (ii) The sheaf (with respect to the fppf topology) associated to the presheaf T → {isomorphism classes in PicG } X/B is representable by a separated algebraic space locally of ﬁnite presentation over B . 62 ABRAMOVICH, OLSSON, AND VISTOLI Proof. This follows for example by the same argument used in [5, Appendix]. ♠ C.28. There is a natural morphism (C.28.1) PicG → PicX/B , L → L⊗n . X/B Denote by PicG [n] the inverse image of the identity e = [OX ] ∈ PicX/B . X/B Another description of this space is as follows. The map C.28.1 lifts naturally to a morphism ⊗n . π ∗ : PicG X/B → PicX/B , L → L Let e : B → PicX/B be the morphism corresponding to OX , and set PicG [n] := PicG ×PicX/B ,e B. X/B X/B The stack PicG [n] classiﬁes pairs (L, ι), where L is a G –twisted sheaf X/B on X and ι : L⊗n ≃ OX is an isomorphism. The space PicG [n] is the X/B coarse moduli space of PicG [n] via the map sending (L, ι) to the class of X/B L. In fact, PicG [n] is a µn –gerbe over PicG [n]. Indeed any point P of X/S X/B PicG [n](B ) can fppf–locally on B be represented by a G –twisted sheaf L X/B with L⊗n ≃ OX . Conversely for any two pairs (L, ι) and (L′ , ι′ ) deﬁning the same point of PicG [n], the two G –twisted sheaves L and L′ become X/B isomorphic after making an fppf base change on B . Then ι and ι′ diﬀer by a ∗ section of f∗ OG ≃ Gm . After making another fppf base change on B so that this unit becomes an n–th power, the two pairs (L, ι) and (L′ , ι′ ) become isomorphic. Moreover, this isomorphism is unique up to multiplication by an element of µn (B ). Proposition C.29. The algebraic space PicG [n] is of ﬁnite type over B . X/B Proof. To prove this statement it suﬃces by a standard limit argument to consider the case when B is a noetherian scheme. In this case by noetherian induction it suﬃces to ﬁnd a dominant morphism B ′ → B such that the ′ restriction of PicG X/B to B is of ﬁnite type. We may therefore also assume that B is a noetherian aﬃne integral scheme. By Chow’s lemma there exists a proper surjection P → G with P an algebraic space. After shrinking on B we may assume that P is also ﬂat over B . Then G ×X P is trivial and by C.15 the pullback map Sec(G /X ) ≃ PicG [n] → PicG X P/P [n] ≃ Sec(G ×X P/P ) G× X/B is of ﬁnite type. Consequently the map PicG [n] → PicG [n] P/B X/B is also of ﬁnite type. It therefore suﬃces to show PicG [n] is of ﬁnite type. P/B Now in the case when G = Bµn × X there exists a globally deﬁned G twisted sheaf L with L⊗n trivial. Choose one such sheaf L. If p : G → X TWISTED YET TAME 63 denotes the projection, then for any invertible sheaf M on X with M⊗n ≃ OX , the sheaf L ⊗ p∗ M is also a G –twisted sheaf. Conversely, if N is a G –twisted sheaf with N ⊗n trivial, then L ⊗ N −1 descends uniquely to an n–torsion sheaf on X . Using this one sees that PicG [n] is a trivial X/B torsor under PicX/B [n]. The proposition therefore follows from [22, 6.27 and 6.28]. ♠ C.30. Since Sec(G /X ) ≃ PicG [n] is a µn –gerbe over PicG [n] this comX/B X/B pletes the proof of C.3 in the special case of C.24 and hence also the proof in general. ♠ Proof of C.4 C.31. Algebraicity of Hom(X , Y ). By the case of algebraic spaces, the stack Hom(X, Y ) is an algebraic space locally of ﬁnite presentation over B . To prove C.4 it therefore suﬃces, as in the proof of C.3, to show that for any morphism f : X → Y the stack Sec(G /X ) is algebraic locally of ﬁnite presentation over B , where G denotes the pullback of Y along the morphism X // X f // Y. This is shown exactly as in [30, 5.35.8] by verifying Artin’s conditions. The key point is Grothendieck’s existence theorem for Artin stacks which is shown in [30, A.1]. Note that this implies that if X /B is a proper ﬂat tame stack and G → X is a morphism with G a tame stack of ﬁnite presentation over B , then Sec(G /X ) is an algebraic stack locally of ﬁnite presentation over B , as it is equal to the ﬁber product of the diagram Hom(X , G )
idX S // Hom(X , X ). C.32. The diagonal of Sec(G /X ) is quasicompact and separated. Lemma C.33. If X /B is a proper and ﬂat tame stack, and I → X is a ﬁnite morphism, then Sec(I/X ) is separated and of ﬁnite type over B . Proof. For the quasicompactness of the diagonal, it suﬃces to show that given two sections s1 , s2 : X → I , the ﬁber product M of the diagram S
s 1 ×s 2 ∆ Sec(I/X ) // Sec(I/X ) ×S Sec(I/X ) is quasicompact. For this we may assume that S is reduced, and furthermore by noetherian induction it suﬃces to show that the map M → S is 64 ABRAMOVICH, OLSSON, AND VISTOLI quasicompact over some dense open in S . Let X0 denote the ﬁber product of the diagram X
s 1 ×s 2 // I ×X I. I Since I is ﬁnite over X , the diagonal of I/X is a closed immersion which implies that the projection j : X0 → X is also a closed immersion. The S space M represents the functor which to any scheme T /S associates the unital set if the base change
X0,T → XT is an isomorphism, and the empty set otherwise. Now to prove that M is of ﬁnite type over S , we may by shrinking if necessary also assume that X0 is ﬂat over S . In this case, if Z ⊂ X denotes the support of the ideal sheaf of X0 , then M is represented by the complement of the closed (since X /S is proper) image of Z in S . This proves the quasicompactness of the diagonal of Sec(I/X ). To verify that the diagonal of Sec(I/X ) is proper, one veriﬁes the valuative criterion using the same argument as in the proof of C.13. The proof that Sec(I/X ) is of ﬁnite type over B now proceeds exactly as in the proof of C.13. ♠ Now let X and Y be ﬁnite type tame B stacks with X proper and ﬂat over B , and let X and Y be the coarse moduli space of X and Y respectively. Fix a morphism f : X → Y , and let G → X be the pullback of Y along the composite X ∆ // X f // Y. Lemma C.34. The diagonal of Sec(G /X ) is quasicompact and separated. Proof. Let s1 , s2 : X → G be two sections, and let I denote the ﬁber product of the diagram (which is a ﬁnite stack over X since G has ﬁnite diagonal) X s ×s
1 2 G − − → G ×X G . −− Then the ﬁber product of the diagram B ∆ s 1 ×s 2 Sec(G /X ) ∆ // Sec(G /X ) ×B Sec(G /X ) is isomorphic to Sec(I/X ) which is quasicompact and separated by C.33. ♠ TWISTED YET TAME 65 C.35. Proposition C.15 still holds. Now an examination of the proof of C.15, following [35, 4.13], shows that once we know that in the case when X is tame the stack Sec(G /X ) is locally of ﬁnite type with separated and quasicompact diagonal, then the proof of C.15 carries over also in the case when X does not necessarily admit a ﬁnite ﬂat surjection from an algebraic space but instead is tame. C.36. Completion of proof. To prove that Sec(G /X ) is quasicompact, it suﬃces by noetherian induction to exhibit a dense open subset of B where this is so. Furthermore, we may without loss of generality assume that B is integral. By Chow’s lemma for Artin stacks [33, 1.1] there exists a proper surjection X ′ → X with X ′ an algebraic space. After shrinking on B we may assume that X ′ is ﬂat over B . Applying C.15 to the stack Sec(G /X ) and using the case when X is an algebraic space we get that Sec(G /X ) is quasicompact. This completes the proof of C.4. References
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