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Unformatted text preview: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 8, Pages 3031–3057 S 0002-9947(02)03011-8 Article electronically published on April 3, 2002 BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES IGOR DOLGACHEV AND JONGHAE KEUM Abstract. We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice II1,25 of rank 26 and signature (1, 25). The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kond¯. We shall explain how our generators are related to the generators o of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface. 0. Introduction Let S : F (x0 , x1 , x2 , x3 ) = 0 be a nonsingular cubic surface in P3 over C. Its Hessian surface is a quartic surface defined by the determinant of the matrix of second-order partial derivatives of the polynomial F . When F is general enough, the quartic H is irreducible and has 10 nodes. It contains also 10 lines which are the intersection lines of five planes in general linear position. The union of these five planes is classically known as the Sylvester pentahedron of S . The equation of S can be written as the sum of cubes of some linear forms defining the five planes. ˜ A nonsingular model of H is a K3-surface H . Its Picard number ρ satisfies the inequality ρ ≥ 16. In this paper we give an explicit description of the group Bir(H ) of birational isomorphisms of H when S is general enough so that ρ = 16. Although H , in general, does not have any non-trivial automorphisms (because S does not), ˜ the group Bir(H ) ∼ Aut(H ) is infinite. We show that it is generated by the = automorphisms defined by projections from the nodes of H , a birational involution which interchanges the nodes and the lines, and the inversion automorphisms of ˜ some elliptic pencils on H . This can be compared with the known structure of the group of automorphisms of the Kummer surface associated to the Jacobian abelian surface of a general curve of genus 2 (see [Ke2], [Ko]). The latter surface is birationally isomorphic to the Hessian H of a cubic surface ([Hu1]), but the Picard ˜ number of H is equal to 17 instead of 16. We use the method for computing Bir(H ) employed by Kond¯ in [Ko]. We show that the Picard lattice SH of the K3-surface o ˜ H can be primitively embedded into the unimodular lattice L = Λ ⊥ U of signature Received by the editors June 30, 2001 and, in revised form, August 27, 2001. 2000 Mathematics Subject Classification. Primary 14J28, 14J50; Secondary 11H56. Key words and phrases. Quartic Hessian surfaces, automorphisms, K3 surface, Leech lattice. Research of the first author was partially supported by NSF grant DMS 9970460. Research of the second author was supported by Korea Research Foundation Grant KRF-2000041-D00014. c 2002 American Mathematical Society 3031 3032 IGOR DOLGACHEV AND JONGHAE KEUM (1, 25), where Λ is the negative definite Leech lattice and U is the hyperbolic plane. The orthogonal complement of SH in L is a primitive lattice of rank 10 which contains a negative definite root lattice of type A5 + A5 . The fundamental domain 1 D of the reflection group of L cuts out a finite polyhedron D in the fundamental domain DH for the (−2)-reflection group of SH in the connected component of the set {x ∈ SH ⊗ R : (x, x) > 0} containing an ample divisor class. We determine the hyperplanes h which bound D and match them with automorphisms h → σh of ˜ H such that σh sends one of the half-spaces defined by h to the other half-space − defined by h or to one of the two half-spaces corresponding to σh 1 . This allows one to prove that the automorphisms σh generate a group of symmetries of DH , having D as its fundamental domain. By the Torelli Theorem for K3-surfaces this implies that the automorphisms σh together with some symmetries of D generate ˜ the group Aut(H ). The reason why the beautiful combinatorics of the Leech lattice plays a role in the description of the automorphisms of H is still unclear to us. We hope that the classification of all K3-surfaces whose Picard lattice is isomorphic to the orthogonal complement of a root sublattice of L will shed more light on this question (for such K3 surfaces, also see [KK]). 1. Some classical facts Here we summarize without proofs some of the known properties of the quartic Hessian surface H of a cubic surface F in P3 . We refer for the proofs to the classical literature (for example, [Ba], [Sa]). (1) H is the locus of points x ∈ P3 such that the polar quadric Px (F ) of F is singular. (2) H is the locus of points x ∈ P3 such that there exists a polar quadric Py (F ) of F such that x ∈ Sing(Py (F )). (3) If F is nonsingular and general enough, then H has 10 nodes corresponding to polar quadrics of corank 2 and has 10 lines corresponding to their singular lines. (4) The lines (resp. singular points) of H are the intersection lines of 10 pairs (resp. 10 triples) of hyperplanes πi , i = 1, . . . , 5, any four of them being linearly independent. The union of the planes πi is called the Sylvester pentahedron of the cubic surface. (5) If li = 0 are the equations of the hyperplanes πi , the equation of F can be 3 3 written in the Sylvester form a1 l1 + . . . + a5 l5 = 0. The equation of H can be 1 1 written in the form a1 l1 + . . . + a5 l5 = 0. (6) H ∩F is the parabolic curve of F : the set of points x ∈ F such that the tangent hyperplane Tx (F ) intersects F along a cubic curve which has a cuspidal point at x. (7) If x ∈ H and y ∈ Sing(Px (F )), then the correspondence x → y is a birational ˜ involution τ of H . On a minimal nonsingular model H it interchanges the exceptional curve blown up from the node Pijk = πi ∩ πj ∩ πk with the line ˜ Llm = πl ∩ πm , where {i, j, k, l, m} = {1, 2, 3, 4, 5}. The involution τ of H is ˜ (τ ) is an Enriques surface. The pair fixed-point-free and its quotient Y = H/ of points in the involution defines a line in P3 . The set of such lines forms the Reye congruence of lines isomorphic to Y (see [Cos]). BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3033 (8) The pencil of planes passing through a line Lij of H cuts out a pencil of cubic curves on H . The plane touching H along the line Lij defines a conic Cij on H. (9) The union of the conics corresponding to the four lines lying in the same plane πi is cut out by a quadric. (10) Projecting from a node Pijk , we get a representation of H as the double cover of P2 branched along the union of two cubics. The two cubics intersect at the six vertices of a complete quadrilateral xyz (x + y + z ) = 0, tangent at three vertices (0, 0, 1), (0, 1, 0), (1, 0, 0) with the tangent directions such that there exists a conic intersecting each cubic only at these three points. 2. The Reye congruence The Hessian surface H is a special case of a quartic symmetroid, that is, a quartic surface given by vanishing of the determinant of a symmetric matrix with linear homogeneous polynomials as its entries. We refer to [Cos] for general properties of quartic symmetroids and the associated Reye congruences of lines. ˜ Let us look closer at the Enriques surface Y = H/(τ ). We have a natural indexing of faces πi of the Sylvester pentahedron by the set {1, 2, 3, 4, 5}, edges by subsets α = {i, j } ⊂ {1, 2, 3, 4, 5}, and the vertices by subsets β = {i, j, k } ⊂ {1, 2, 3, 4, 5}. A vertex Pijk belongs to an edge Lmn if and only if {i, j, k } ⊃ {m, n}. Let Nα be ˜ the smooth rational curve on H corresponding to Pα and similarly let Tβ be the smooth rational curve corresponding to Lβ . The Enriques involution τ interchanges the curves Nα and Tβ , where α and β are complementary sets. Let Uβ be the image on Y of the pair (Nα , Tβ ). We have Uab · Ucd = 1, if {c, d} ∩ {a, b} = ∅; . 0, otherwise. The dual intersection graph is the famous Petersen tri-valent graph with group of symmetries isomorphic to the permutation group S5 . U2 3 U4 1 U4 5 U U3 5 U U1 5 U2 4 13 U5 2 U4 3 12 Figure 1. The sum ∆ of the nodal curves Uab satisfies ∆2 = 10. It defines an embedding of Y into the Grassmann variety G(2, 4) which exhibits Y as the Reye congruence of lines for the web of polar quadrics to F . 3034 IGOR DOLGACHEV AND JONGHAE KEUM Let D12 = U12 + U34 + U35 + U45 and F12 = ∆ − D12 . Then |F12 | is an elliptic pencil on Y with degenerate fibre F12 of Kodaira type I6 and one fibre of type I2 formed by U12 and the image of the conic C12 . Similarly we define other elliptic pencils |Fab |. There are ten such elliptic pencils on Y . We have Fab · Fcd = 4 if {c, d} = {a, b}. Note that Fab = 6∆, ab Eab = 3∆, ab where 2Eab is a multiple fibre of |Fab |. If Uab and Ucd intersect, the sum Dab + Dcd ∗ ˜ is a degenerate fibre of type I2 = D5 of the elliptic pencil |2∆ − Fab − Fcd | = |Dab + Dcd |. The linear system |Fab + Fcd | defines a finite map of degree 2 from Y to a Del Pezzo surface D4 of degree 4. It blows down two Dynkin curves of type A2 . The ramification curve is of genus 3. ˜ The pull-backs of the elliptic pencils on Y to H give elliptic pencils of two types: ∗ Type I with degenerate fibres 2I6 + 2I2 and Type II with degenerate fibres 2I2 . A pencil of Type I is cut out by planes through a line Lij or, in the double plane construction, by the pencil of cubic curves spanned by the components of the branch locus. 3. The Picard lattice Let us consider the double plane construction of H and the corresponding elliptic ˜ ˜ pencil. The surface H admits a double cover π : H → V to a rational elliptic surface V obtained by blowing up the base points of the pencil of cubic curves spanned by the components of the branch locus. It is branched along two smooth elliptic fibres. The elliptic fibration f : V → P1 has one reducible fibre of type I6 and one of type I2 . Let Ei , i = 0, . . . , 5, be the irreducible components of the fibre of type I6 , and let E6 be one of the irreducible components of the fibre of type I2 . ˜ 3.1. Lemma. The Picard group Pic(H ) is generated by π ∗ (Pic(V )) and the divisor classes of the curves Ei , i = 0, . . . , 6. ˜ Proof. Consider the pull-back f : H → P1 of the elliptic fibration f with the base obtained by the double cover of P1 ramified at two points. It has two reducible fibres of type I6 and two of type I2 . The Mordell-Weil group MW(f ) of f is of rank 2 and is isomorphic to the Mordell-Weil group MW(f ) of f . This can be seen as ˜ follows. Since the Picard number of H is equal to 16, the rank of MW(f ) is equal to 16 − 10 − 2 − 2 = 2. Let us choose the zero section of f to be the pre-image of the ˜ zero section of f . Then the covering involution α of the double cover π : H → V ∗ defines an automorphism of the group MW(f ) which is identical on π (MW(f )). Suppose we have a section s ∈ MW(f ) \ π ∗ (MW(f )). Since MW(f )/π ∗ (MW(f )) is a group of finite order, there exists an integer n such that ns ∈ π ∗ (MW(f )) and hence nα(s) = α(ns) = ns. This implies that the section α(s) − s is of finite order and intersects the zero section at the points of the ramification divisor of π . However, it is well known that any two different torsion sections of an elliptic fibration are disjoint. It is known that the Picard group of an elliptic surface is generated by sections ˜ and components of fibres. By the previous argument, any section on H is a pre5 image of a section on V . The sum i=0 Ei is a fibre of f . Since Ei + α(Ei ) is the ˜ pre-image of a component of a fibre of f , we obtain that any divisor class on H can BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3035 be written as a sum D1 + D2 , where D1 is the pre-image of a divisor class on V and D2 is a linear combination of the divisor classes of the curves Ei , i = 0, . . . , 6. This proves the lemma. ˜ 3.2. Corollary. The Picard group Pic(H ) is generated over integers by the twenty smooth rational curves Nα and Tβ . ˜ Proof. We have seen already that Pic(H ) is generated by π ∗ (Pic(V )) and the curves Ei , i = 0, . . . , 6. The latter curves belong to the set N T = {Nα , Tβ }. The group Pic(V ) is generated by the exceptional curves blown up from the vertices of the quadrilateral and the proper transforms of its sides. The pre-images of all these ˜ curves in H belong to the set N T . ˜ Let TH be the lattice of transcendental cycles on H . By definition, it is equal ˜ ˜ ˜ ) in the cohomology lattice L = H 2 (H, Z). to the orthogonal complement of Pic(H ˜ Since the latter is a unimodular lattice, the lattices Pic(H ) and TH have isomorphic ˜ discriminant groups, and the quadratic forms on the discriminant groups differ only ˜ by the sign. We are not able to give a formula for Pic(H ) in terms of an orthogonal sum of some standard lattices. However, it is possible to give such a formula for the lattice TH . ˜ Let An denote the negative definite root lattice of type An (it is defined by the matrix equal to the Cartan matrix of the root system of type An taken with the negative sign), let U be the standard hyperbolic plane, and, for any lattice M , let M (m) denote the lattice M with quadratic form multiplied by m. 3.3. Theorem. TH ∼ U ⊥ U (2) ⊥ A2 (2). ˜= ˜ Proof. Consider the sublattice S of Pic(H ) spanned by π ∗ (Pic(V )) and the divisor ∗ ∗ classes Ei = [Ei − α(Ei )], i = 0, . . . , 6. We have 5=0 Ei = 0. This gives i S ∼ π ∗ (Pic(V )) ⊥ = 6 ∗ Z[Ei ] ∼ A1 (−1) ⊥ A⊥9 ⊥ A5 (2) ⊥ A1 (2). = 1 i=1 Note that 2[Ei ] = ([Ei + α(Ei )]) + ([Ei − α(Ei )]) ∈ S. ˜ Obviously Pic(H )/S is generated by the classes [Ei ] of the divisors Ei , i = 0, . . . , 6, which are elements of order 2. Also these classes are linearly independent. In fact, assume that for some subset I ⊂ {0, . . . , 6} we have eI = i∈I [Ei ] ∈ S . Write eI ∗ as a sum π ∗ (x) + y , where x ∈ Pic(V ), y ∈ 6=0 Z[Ei ]. Then i [Ei + α(Ei )] = π ∗ (2x). eI + α(eI ) = i∈I However, Ei + α(Ei ) = π ∗ (Ei ), where Ei is a component of a fibre of the elliptic fibration f . It is easy to see that the divisor class of the sum i∈I Ei is not divisible by 2 in Pic(V ). Thus we have ˜ Pic(H )/S ∼ (Z/2)7 . = ˜ This shows that the discriminant of the lattice Pic(H ) is equal to 218 · 3/214 = 24 · 3. Moreover, let us show that the discriminant quadratic form qD of the discriminant 3036 IGOR DOLGACHEV AND JONGHAE KEUM ˜ group D of Pic(H ) coincides with minus the discriminant quadratic form of the lattice U ⊥ U (2) ⊥ A2 (2). We have ˜ ˜ S ⊂ Pic(H ) ⊂ Pic(H )∗ ⊂ S ∗ . ˜ This shows that the group A = Pic(H )/S is a subgroup of the discriminant group D(S ) of S which is isotropic with respect to the discriminant quadratic form. The discriminant group D is equal to A⊥ /A, where the orthogonal complement means the orthogonal complement with respect to the quadratic form of D(S ) (see [Ni]). Let s0 , E1 , s1 , s2 , E3 , s3 , s4 , E5 , s5 be the 9 curves on V coming from blowing up of P2 . Here, E1 , E3 , E5 are (−2)-curves contained in the reducible fibre of f of type I6 , and s0 , s1 , . . . , s5 are (−1)-curves which are sections of f . Then Pic(V ) is generated by these nine curves and the full transform l0 of a line on P2 . The 10 divisor classes e0 = π ∗ (l0 ), e1 = π ∗ (E1 + s1 ), e2 = π ∗ (s1 ), e3 = π ∗ (E3 + s3 ), e4 = π ∗ (s3 ), e5 = π ∗ (E5 + s5 ), e6 = π ∗ (s5 ), e7 = π ∗ (s0 ), e8 = π ∗ (s2 ), e9 = π ∗ (s4 ), form a standard orthogonal basis of π ∗ (Pic(V )) ∼ A1 (−1) ⊥ A⊥9 . The lattice = 1 ∗ S is generated by these 10 classes and 6 divisor classes Ei , i = 1, . . . , 6. The discriminant group D(S ) is generated by the cosets of ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ e0 /2, . . . , e9 /2, E2 /2, . . . , E5 /2, (E1 + 2E2 + 3E3 + 4E4 + 5E5 )/12, E6 /4. ˜ We can easily express the generators [Ei ], i = 0, . . . , 6, of the group A = Pic(H )/S in terms of these generators; ∗ [E1 ] = [E1 + α(E1 )]/2 + [E1 − α(E1 )]/2 = π ∗ (E1 )/2 + E1 /2 ∗ = (e1 − e2 )/2 + E1 /2. Similarly, ∗ [E2 ] = (e0 − e1 − e3 − e8 + E2 )/2, ∗ [E3 ] = (e3 − e4 + E3 )/2, ∗ [E4 ] = (e0 − e3 − e5 − e9 + E4 )/2, ∗ [E5 ] = (e5 − e6 + E5 )/2, 5 ∗ [E6 ] = (e0 − e7 − e8 − e9 + E6 )/2, 9 [Ei ] = (3e0 − i=0 ei )/2. i=1 Now it is straightforward to compute the group A⊥ /A and its quadratic form. Indeed, it is generated by four elements ∗ ∗ ∗ ∗ ∗ x = (E1 + 2E2 + 3E3 + 4E4 + 5E5 )/6, ∗ u = (e7 + e8 + E1 )/2, ∗ y = E6 /2, ∗ v = (e1 + e3 + e5 + e6 + E2 )/2, and its quadratic form is isomorphic to x + y + v, y + u ⊕ u, y + v ∼ −qA2 (2)⊥U (2) . = Applying Nikulin’s results [Ni], we obtain that there exists a unique (up to isomorphism) lattice of signature (2, 4) with the discriminant quadratic form isomorphic ˜ to minus the discriminant quadratic form of Pic(H ). Hence, the transcendental ˜ lattice of H is isomorphic to U ⊥ U (2) ⊥ A2 (2). This proves the theorem. BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3037 3.4. Remark. By a different method (using a computer) the previous result was independently obtained by B. van Geemen [vGe]. We set N= Nα , T= α Tβ . β ˜ Let ηH be the pre-image on H of the class of a hyperplane section of H and let ηS be its image under the Enriques involuton τ . It is known (see, for example, [Cos], Proposition 2.4.1) that (3.1) 2ηH = 3ηS − T , 2ηS = 3ηH − N . ˜ In particular, the pre-image ∆ of the class ∆ on the Enriques surface can be expressed as ˜ (3.2) ∆ = T + N = ηH + ηS . Pick up a face of the Sylvester pentahedron, say π5 . It is immediately checked that 2(ηS − ηH ) intersects each Nα and Tβ with the same multiplicity as the divisor (T15 + T25 + T35 + T45 ) − (N123 + N124 + N134 + N234 ). This gives a linear relation: 2(ηS − ηH ) = (T15 + T25 + T35 + T45 ) − (N123 + N124 + N134 + N234 ). Denote the first bracket by T5 and the second bracket by N5 . Similarly introduce Ti , Ni , i = 1, . . . , 4, for any other face. We obtain (3.3) 2(ηS − ηH ) = Ti − Ni , i = 1, . . . , 5. One can show that the linear system |ηH + 1 (Ti − Ni )| = |ηS | defines a standard 2 cubic Cremona involution τi with fundamental points at the vertices of the tetrahedron formed by the faces πj , j = i. All these involutions restrict to the Enriques ˜ involution on H (cf. [Hu2], p.335). ˜ We shall denote by Cij the pre-image on H of the conic cut out by the plane which is tangent to H along the edge Lij . Its divisor class equals (3.4) Cij = ηH − 2Tij − Nijk − Nijl − Nijm . ˜ We shall also denote by Rklm the pre-image on H of the cubic cut out by the plane through the edge Lij and the opposite vertex Pklm . Its divisor class is equal to (3.5) Rklm = ηH − Tij − Nijk − Nijl − Nijm − Nklm . One can show ([Hu2]) that any rational nonsingular curve of degree ≤ 3 is one of the described above (on a general H ). 4. Kummer surfaces and Hessians Let C be a genus 2 curve and (p1 , . . . , p6 ) its ordered set of Weierstrass points. Then the divisor classes µ0 = 0, µij = [pi + pj − 2p6 ], 1 ≤ i < j ≤ 6, are the sixteen 2-torsion points on its Jacobian variety J . In the usual way we identify the group of 2-torsion points J2 with the set of 2-element subsets α of the set S = {1, 2, 3, 4, 5, 6}; the zero point corresponds to the empty subset. We shall identify a 2-element subset α with its complementary subset S \ α. Then the addition of points corresponds to the symmetric sum α + β of subsets. One defines the symplectic bilinear form with values in F2 on J2 by (4.1) (µα , µβ ) = |α ∩ β | modulo 2. 3038 IGOR DOLGACHEV AND JONGHAE KEUM Let us write elements of the vector space V = F4 as 2 × 2 matrices [ η ] with rows 2 in F2 . We define the quadratic form q0 : V → F2 by q0 ([ η ]) = · η, where the dot 2 means the standard dot-product in F2 . The associated symmetric bilinear form is 2 ( η , η )= ·η +η· . It is a non-degenerate symplectic form. We define an isomorphism of symplectic spaces ψ : J2 → V by µ12 → 10 , 00 µ34 → 01 , 00 µ16 → 00 , 10 µ45 → 00 . 01 Let us identify the span U1 of µ12 , µ16 with the set [4] = {1, 2, 3, 4} by assigning 0 to 1, µ16 to 2, µ12 to 3 and µ12 + µ16 = µ26 to 4. Similarly we identify the span U2 of µ34 , µ45 with [4] by assigning 0 to 1, µ45 to 2, µ34 to 3 and µ34 + µ45 = µ35 to 4. Each 2-torsion point can be written uniquely as the sum a + b, a ∈ U1 , b ∈ U2 , and hence can be identified with the pair (a, b) ∈ [4] × [4], or, equivalently, with a dot in the following 4 × 4 table: • • • • • • • • • • • • • • • • Let C → J, x → [x − p6 ], be the Abel-Jacobi map corresponding to the point p6 . The image of C is denoted by Θ0 . Let Θij = Θ0 + µij be its translate by a 2-torsion point µij . Each Θij contains exactly six 2-torsion points, namely the points µij , µk6 + µij , k = 1, . . . , 5. In other words, µα ∈ Θβ ⇔ β + α ∈ {∅, {16}, {26}, {36}, {46}, {56}}. One also employs different indexing of theta divisors Θα . For each α there exists a partition of [6] = {1, 2, 3, 4, 5, 6} into two disjoint subsets S ∪ S with odd numbers of elements. For α = k 6, it is uniquely determined by the property µβ ∈ Θα if and only if β ⊂ S or β ⊂ S . For β = k 6, we simply take S = {k }. We use either S or S for the index. In this correspondence, Θβ corresponds to Θβ +{6} . For example, Θ12 corresponds to Θ345 or Θ126 , and Θ16 to Θ1 or Θ23456 . Yet there is another classical notation for a theta divisor. Each theta divisor is equal to the set of zeroes of a theta function θ [ ] (z, τ ) with theta characteristic [ ]. The theta characteristic corresponding to ΘS , where #S is odd, is equal to ψ (S + {135}). For example, the theta characteristic corresponding to Θ12 = Θ345 is ψ (14) = [ 1 1 ], where and 11 are the first and the second rows of the matrix. One assigns to Θα a dot (ab) ∈ [4] × [4] in the 4 × 4 table as above in such a way that (ab) in the right-side table contains a 2-torsion point corresponding to the entry (cd) in the left-side table if and only if a = c or b = d but (ab) = (cd). Explicitly, we have two tables µ0 µ16 µ12 µ26 µ45 µ23 µ36 µ13 µ34 µ25 µ56 µ15 µ35 µ24 µ46 µ14 Θ12 Θ26 Θ0 Θ16 Θ36 Θ13 Θ45 Θ23 Θ56 Θ15 Θ34 Θ25 Θ46 Θ14 Θ35 Θ24 The following construction of Hutchinson [Hu1] describes the translation of the left-hand-side table to the table of the corresponding values of the map ψ , and, at BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3039 the same time, translates the right-hand-side table to the table of the corresponding theta characteristics. Denote the four columns 1100 1010 by the numbers 1, 2, 3, 4, respectively. Then the theta characteristic of the (ab)entry in the right-hand-side table is equal to the characteristic formed by the columns (ab). Now, if we reverse the order of the 4 vectors and do the same, we obtain the values of ψ at the entries of the left-hand-side table. o An affine plane in J2 is called an odd (or G¨pel) tetrad if it is a translation of a totally isotropic linear subspace of dimension 2. Otherwise it is called an even (or Rosenhain) tetrad. There are 60 odd and 80 even tetrads, each set forming an orbit with respect to the group generated by symplectic automorphisms of J2 and translations. For example, the rows and columns of the left-hand-side table above correspond to even tetrads but the diagonal corresponds to an odd tetrad. A Weber hexad is defined as the symmetric sum of an even and an odd tetrad which have one point in common. For example, the dots marked with asterisk represent a Weber hexad: ∗ • • • • ∗ • • ∗ ∗ • ∗ • • • ∗ A Weber hexad has the following property. Each theta-divisor contains either 3 points from a Weber hexad or just one. The number of theta divisors which contain three points is equal to ten. One can choose 5 subsets of four elements from this set of ten theta divisors such that each point from the Weber hexad is contained in two theta divisors from this set. For the Weber hexad chosen above, the five sets are the following: A1 = (Θ56 , Θ46 , Θ15 , Θ14 ), A2 = (Θ14 , Θ36 , Θ16 , Θ34 ), A3 = (Θ23 , Θ25 , Θ56 , Θ36 ), (4.2) A4 = (Θ23 , Θ26 , Θ34 , Θ46 ), A5 = (Θ26 , Θ16 , Θ25 , Θ15 ). By inspection, one sees that the theta characteristics of theta functions from the same group of four add up to zero. This means that the sum of the divisors belongs to the linear system |4Θ|, where Θ = Θ24 has the zero theta characteristic. Thus we have five divisors D1 , . . . , D5 ∈ |4Θ|, each passing through every point of the Weber hexad with multiplicity 2. Let θi , i = 0, . . . , 4, be the corresponding theta functions of order 4 with zero theta characteristic. They define a rational map Φ : J → P4 . 4.1. Theorem. The image H of Φ is contained in a linear hyperplane x4 = ax0 + bx1 + cx2 + dx3 , where a, b, c, d = 0, and satisfies the quartic equation ax1 x2 x3 x4 + bx0 x2 x3 x4 + cx0 x1 x3 x4 + dx0 x1 x2 x4 + x0 x1 x2 x3 = 0. In particular, H is isomorphic to the Hessian of a nonsingular cubic surface. We refer for the proof to [Hu1] or [vGe]. One can also find there the expression of the coefficients a, b, c, d in terms of theta constants. 3040 IGOR DOLGACHEV AND JONGHAE KEUM Since Θ is a symmetric theta divisor, the rational map Φ factors through the map J → Kum(J ) ⊂ P3 given by the linear system |2Θ|. The image of the Weber hexad in the Kummer surface is a set of 6 nodes, a Weber set of nodes. The linear system |4Θ| is equal to the inverse transform of a linear system |Lh | of quadrics through the Weber set of nodes h. Thus the Hessian surface H = Im(Φ) is the ¯ birational image of Kum(J ) under the map Φ given by the linear system |Lh |. An explicit equation of the locus of the cubic surfaces whose Hessian is birationally isomorphic to a Kummer surface inside of the moduli space of cubic surfaces has been found in [Ro], [vGe]. Let us choose the Weber hexad as above: (4.3) h = (µ0 , µ23 , µ34 , µ25 , µ15 , µ14 ). Let (4.4) (Θ56 , Θ46 , Θ15 , Θ14 , Θ36 , Θ16 , Θ34 , Θ23 , Θ25 , Θ26 ) be the corresponding set of ten theta divisors which contain exactly three of the points from h and hence three points from the set J2 \ h. The images of each node from h under the map Φ form a conic in H . The plane section of H along such a conic is equal to the union of two conics. These conics do not appear on a general Hessian. The images of the remaining nodes which are the images of ten 2-torsion points µα are ten nodes of the Hessian which we denote by Nα . The image of a theta divisor Θβ from (4.4) is a line on H which contains three nodes Nα such that µα ∈ Θβ . We denote this line by Tβ . Note that each Θβ belongs to exactly two of the five subsets from (4.2), say Ai , Aj . If we reindex Tβ by Tij and Nα = Nijk , where Nα ∈ Tij , Tjk , Tik , then we get the notation used for the nodes and the lines on a general Hessian. The image of a theta divisor not belonging to the group of ten is a rational curve of degree 3 on H which passes through 5 nodes. These curves do not appear on a general Hessian. 4.2. Remark. A Weber set h of nodes of a Jacobian Kummer surface K is a set P of 6 points in P3 in general linear position. Recall that each such set defines the Weddle quartic surface W (P ) which is the locus of singular points of quadrics passing through the set P . The image of W (P ) by the linear system of quadrics through P is a Kummer surface K (P ) birationally isomorphic to W (P ). The image of K is a Hessian quartic H . The quartic surfaces K (P ) and H touch each other along a curve of degree 8 not passing through their nodes. There are 192 Weber hexads. The affine symplectic group 24 Sp(4, F2 ) of order 24 · 6! acts transitively on the set of Weber hexads. The isotropy group of a Weber hexad is isomorphic to the alternating group A5 . Let Mw be the moduli space of Jacobian Kummer Kum surfaces together with a choice of a Weber hexad of nodes. It is a cover of degree 12 of the moduli space M2 of genus 2 curves isomorphic to the moduli space MKum of Jacobian Kummer surfaces. The above construction defines a map Mw Kum → MKum. What is the degree of this map? 4.3. Remark. Since the K3-surface birationally isomorphic to the Hessian quartic of a nonsingular cubic surface admits an Enriques involution, any K3-surface birationally isomorphic to a Jacobian Kummer surface admits an Enriques involution. This fact, of course, is known, and it is also true for not necessarily Jacobian BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3041 Kummer surfaces (see [Ke1]). But here we get an explicit construction of this involution. Applying (3.1), we see that the Enriques involution on the Hessian quartic surface associated to a Kummer surface K is given by the linear system defined by the divisor 1 3 1 Ep − Ep , ηS ∼ (3ηH − N ) ∼ 3ηK − 2 2 2 p∈W p∈W where ηK is the class of a hyperplane section of the Kummer surface K in P3 , W is the Weber hexad of nodes corresponding to the Weber hexad of 2-torsion points, and Ep is the class of the exceptional curve blown up from one of the 16 nodes of K . On the abelian surface J this corresponds to the linear system |6Θ − 3 µ∈h µ − µ∈h µ|. The linear system µ− |6Θ − 3 µ∈h µ + (4Θ − 2 µ∈h µ)| = |10Θ − 5 µ∈h µ− µ∈h µ| µ∈h maps J onto an Enriques surface embedded in P5 by its Fano linear system of degree 10. If we choose the Weber hexad (4.3), then the Sylvester pentahedron of the corresponding Hessian quartic is given in Figure 2. We shall use the previous notation for nodes and lines on a general Hessian surface. N N T15 T 24 16 T 46 T T N 36 14 16 T N 26 34 46 T36 T 23 N N 56 T 56 12 N T N 35 45 25 N 13 N 26 Figure 2. 5. Elliptic pencils ˜ There are many elliptic pencils on the surface H . We list a few, which will be used later. We accommodate the notations for the conics Cα (3.4) and cubics Rα (3.5) to our new notation for (−2)-curves Tα and Nβ . 3042 IGOR DOLGACHEV AND JONGHAE KEUM Type 1. It is cut out by the linear pencil of planes through an edge of the pentahedron: |Fα | = |Cα + Tα |. We have Fα · Fβ = 2, if Tα and Tβ are skew. The pencil |Fα | has 2 reducible fibres of type I2 : Cα + Tα , Nα + Rα , ¯ ¯ where Rα is the residual cubic for the plane section of H passing through the edge ¯ Tα and its opposite node Nα (see (3.5)). It has also two reducible fibres of type I6 ¯ corresponding to the two faces containing Tα . Observe that the Enriques involution leaves the pencil invariant by interchanging two degenerate fibres of the same type. The members of the pencil are cubic curves. The pencil of type 1 is denoted by Fi in [CD]. It is the pre-image of one of the ten pencils on the Enriques surface Y as explained in Section 2. Using (3.2), we have ˜ |∆ − 2Fi | = Bi + τ (Bi ), where τ is the Enriques involution, and Bi is of the form T15 + N56 + N24 + N16 . This agrees with the notation from [CD]. ˜ Type 2. This is a pencil |FNα ,Tβ | on H formed by proper transform of quartic elliptic curves cut out by the pencil of quadric cones with the vertex at a node Nα which contain the lines through Nα and tangent along one of them, Tβ . For example, |FN16 ,T15 | = |2ηH − 2T15 − T16 − T14 − N24 − N56 − N12 − N13 − N46 − N35 − 2N16 |. It has one reducible fibre of type I8 : T25 + N26 + T26 + N24 + T46 + N45 + T56 + N56 , and two reducible fibres of type I4 : C15 + N36 + T34 + T36 , C23 + N16 + T16 + T14 . Observe that N46 , N35 , N13 , N12 are sections and T23 , T15 are bisections of the elliptic fibration. Let us take N46 as the zero section. We easily check (by intersecting both sides with any Nα and any Tβ ) that 2N13 − 2N46 ∼ −T36 + T34 + T14 − T16 + T46 N56 − 2T25 − N26 + N24 + 2T46 + N45 . This implies that N13 is a 2-torsion section. Also, N46 + N12 ∼ N13 + N35 − N46 − T34 + T36 − T46 + T56 + N56 + T25 − T26 − N24 implies that N13 ⊕ N35 = N12 , where ⊕ is the group operation on the set of sections with zero section defined by N46 . One also checks that the translation by N13 sends T15 to R36 and T23 to R16 . Note that |FN16 ,T15 | = |FN36 ,T23 |. There are 15 elliptic pencils of type 2. BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3043 ˜ Type 3. This is a pencil |FTβ ,Tγ | on H of proper transforms of quartic elliptic curves which are cut out by the pencil of quadric cones with the vertex at a node Nα and tangent to H along two edges Tβ , Tγ intersecting at Nα . It is spanned by the double plane through the two edges and the union of two planes tangent along these edges. For example, |FT16 ,T14 | = |2ηH − 2T16 − 2T14 − N12 − N13 − N46 − N35 − 2N16 | = |C16 + C14 |. It has reducible fibre of type ∗ I0 ∗ and I2 : C23 + 2T15 + N16 + N56 + N24 , 2T36 + 2T34 + 2N36 + N13 + N12 + N46 + N35 . Also it has a reducible fibre of type I2 : C16 + C14 . We see that N45 and N26 are contained in fibres. Since C16 + C14 − N26 (resp. C16 + C14 − N45 ) is of degree 4 with respect to ηH , and cannot be a union of two conics, we get two more reducible fibres of type I2 . Observe that T26 , T56 , T25 and T46 are sections and T16 , T14 , T23 are bisections. Let us take T46 as the zero section. Then 2T25 − 2T46 ∼ 2T36 + 3N36 + 4T34 + N35 + 3N46 + 2N12 − N56 + N24 + C16 + N45 − N26 − 2(C16 + C14 ) implies that T25 is a 2-torsion section. Also T25 ⊕ T56 = T26 . 6. Birational involutions of a Hessian quartic There are some obvious birational involutions of a Hessian quartic surface. They are: (i) (ii) (iii) (iv) the Enriques involution τ ; an involution pα defined by projection from a node Nα ; an involution defined by a pair of skew lines; an involution defined by two elliptic pencils |F | and |F | with F · F = 2. This ˜ is the covering involution for a degree 2 map H → P1 × P1 defined by the linear system |F + F |. (v) an involution defined by the inversion map of an elliptic fibration with a section; (vi) an involution defined by the translation by a 2-torsion section in the group law of sections of an elliptic fibration. ˜ Let us describe the action of each involution on the Picard lattice SH of H . (i) We have already described the action of the Enriques involution. In our new notation we have N16 ←→ T23 , N24 ←→ T36 , N56 ←→ T34 , N12 ←→ T56 , N13 ←→ T46 , N26 ←→ T14 , N35 ←→ T26 , N46 ←→ T25 , N36 ←→ T15 , N45 ←→ T16 . We already know that the image of ηH is equal to ηS . 3044 IGOR DOLGACHEV AND JONGHAE KEUM (ii) We may assume that we project to the plane defined by the opposite face of the pentahedron. Then the projection p16 from N16 acts as follows: N56 ←→ N24 , N13 ←→ N12 , N35 ←→ N46 , T25 ←→ T26 , T36 ←→ T34 , T56 ←→ T46 , T23 ←→ ηH − N16 − T23 − N26 − N45 − N36 = R16 , and the remaining curves N26 , N36 , N45 , T15 , T16 , T14 are fixed. By (3.4) and (3.5), N16 + R16 = C23 + T23 , and hence N16 ←→ C23 . Since ηH = C23 + 2T23 + N36 + N45 + N26 , we obtain ηH ←→ N16 + 2R16 + (N36 + N45 + N26 ) = 2ηH − 2T23 − N26 − N45 − N36 − N16 . Thus the involution is given by the linear system of quadrics through the vertex N16 and touching H along the edge T23 . Projections pα commute with the Enriques involution τ , i.e. pα ◦ τ = τ ◦ pα . (iii) This is a special case of (iv) when the two pencils are of type 1 and correspond to two skew edges. Assume that the edges are T15 and T23 . Let the fibres N12 + T16 + N13 + T25 + N26 + T26 and T14 + T46 + N46 + N15 + T56 + N35 of F15 go to the lines A1 , A2 on the quadric P1 × P1 . Similarly, let T56 + T36 + T25 + N56 + N35 + N13 and T26 + T46 + T34 + N12 + N46 + N24 go to the lines B1 , B2 of the other ruling. Then the pre-image of A1 splits in T16 + N26 , the pre-image of A2 splits in T14 + N45 , the pre-image of B1 splits in T36 + N56 , the pre-image of B2 splits in T34 + N24 . This easily shows that the action of the involution is defined as follows: T16 ←→ N26 , T14 ←→ N45 , T36 ←→ N56 , T34 ←→ N24 , T15 ←→ C15 , N36 ←→ R36 , N16 ←→ R16 , T23 ←→ C23 , T25 ←→ N13 , T56 ←→ N35 , T26 ←→ N12 , T46 ←→ N46 . This implies that ηH ←→ T15 + 2C15 + T36 + T34 + R16 . This involution is the same as the composition τ ◦ p16 ◦ p36 . (iv) Let us consider the two pencils defined by planes through non-skew edges. Take the edges T15 and T25 . Computations similar to (iii) show that the involution coincides with the projection from N56 . (v) Consider a pencil of type 3 with reducible fibres C16 + C14 , N26 + N26 , N45 + N45 , C23 + 2T15 + N16 + N56 + N24 , 2T36 + 2T34 + 2N36 + N13 + N12 + N46 + N35 . We verify that T46 , T56 , T26 and T25 are sections. Let us take T46 as the zero section. We have already observed that T25 is a 2-torsion section. Consider the automorphism φ of the surface defined by the inversion map with respect to the group law on the set of sections with zero section defined by T46 . Obviously, T46 and T25 are invariant. Also, the components of reducible fibres which they intersect BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3045 are also invariant. Finally, the multiple components of fibres are invariant. This easily shows that all irreducible components of fibres are invariant. Let M be the sublattice spanned by irreducible components of fibres and the sections T46 , T25 . Its rank is 15. Let α ∈ SH be a primitive vector orthogonal to M . Obviously, φ(α) = −α. One easily finds the vector α: α = 2ηH − 2ηS + 2(T46 + T25 + T34 + T36 + T15 ) + (N36 + N46 + N56 + N13 + N24 ). We leave it to the reader to check that α is a (−6)-root of SH . Therefore, the action of φ on the Picard lattice SH is a reflection φ(x) = x − (6.1) 2(x · α) α. α2 7. The Leech lattice We follow the notation and the main ideas from Kond¯’s paper [Ko]. First we o ˜ embed the Picard lattice SH of H in the lattice L = Λ ⊥ U ∼ II1,25 , where Λ is = the negative definite Leech lattice and U is the hyperbolic plane. We denote each vector x ∈ L by (λ, m, n), where λ ∈ Λ, and x = λ + mf + ng, with f, g being ,λ the standard generators of U . Note that, r = (λ, 1, −1 − λ2 ) satisfies r2 = −2. Such vectors will be called Leech roots. Recall that Λ can be defined as a certain 1 · lattice in R24 = RP (F23 ) equipped with the inner product x, y = − x8y . For any 1 subset A of Ω = P (F23 ) let νA denote the vector i∈A ei , where {e∞ , e0 , . . . , e22 } is the standard basis in R24 . Then Λ is defined as a lattice generated by the vectors νΩ − 4ν∞ and 2νK , where K belongs to the SL(2, F23 )-orbit of the ordered subset (∞, 0, 1, 3, 12, 15, 21, 22) of Ω. These sets form a Steiner system S (5, 8, 24) of eightelement subsets of Ω such that any five-element subset belongs to a unique element of S (5, 8, 24). All such sets are explicitly listed in [To]. ⊥ 7.1. Lemma. There is a primitive embedding of SH in L such that SH contains a sublattice of index 2 isomorphic to the root lattice R = A5 + A5 . 1 Proof. Consider the following Leech roots: x = (4ν∞ + νΩ , 1, 2), y = (4ν0 + νΩ , 1, 2), x0 = (4ν∞ + 4ν0 , 1, 1), xi = (2νKi , 1, 1), z = (0, 1, −1), i = 1, . . . , 5, where K1 = {∞, 0, 1, 2, 3, 5, 14, 17}, K2 = {∞, 0, 1, 2, 4, 13, 16, 22}, K3 = {∞, 0, 1, 2, 6, 7, 19, 21}, K4 = {∞, 0, 1, 2, 8, 11, 12, 18}, K5 = {∞, 0, 1, 2, 9, 10, 15, 20}. It is easy to verify that the inner product of the vectors x, y, z, xi is described by the following (reducible) Coxeter-Dynkin diagram: x • −− z • −− y • x0 • x1 • x2 • x3 • x4 • x5 • 3046 IGOR DOLGACHEV AND JONGHAE KEUM Thus these vectors span a root sublattice R0 of L isomorphic to A3 + A6 . We shall 1 add one more vector to the previous set. Let (7.1) r0 = (2νK0 , 1, 1), K0 = {∞, 1, 2, 3, 4, 6, 15, 18}. One verifies that r0 , y = r0 , x0 = 1, r0 , x = r0 , z = r0 , xi = 0, i = 0. Thus the new Dynkin diagram looks like x • z −− • −− y • −− r0 • −− x0 • x1 • x2 • x3 • x4 • x5 • So, the new lattice R spanned by x, y, z, xi , r0 is isomorphic to A5 + A5 . Let 1 5 θ= 1 (x + y + xi ). 2 i=0 Let qM : D(M ) = M ∗ /M → Q/2Z denote the discriminant quadratic form of a lattice M . We have 5x + 4z + 3y + 2r0 + x0 5 ), qA5 = − , D(A5 ) = ( 6 6 xi 1 ), qA1 = − . 2 2 ¯ Write θ = θ ∈ D(R) as a sum of v = 5x+4z+3y+2r+x0 and βi = xi modulo R. 6 2 ¯ ¯ We have θ = 3v + 5=1 βi . It is checked that θ spans a subgroup A of D(R) i which is isotropic with respect to the discriminant quadratic form qR . Let T be the overlattice of R corresponding to the group A⊥ /A. It is easy to check that ¯ qT = qR |A⊥ /A = v + β1 , v + β2 , v + β3 , v + β4 , v + β5 /θ D(A1 ) = ( = v + β1 + β3 + β5 , β1 + β2 ⊕ β1 + β2 + β3 + β4 , β1 + β2 + β4 + β5 = qA2 (2) ⊕ qU (2) . Thus T = R, θ has the same discriminant quadratic form as the transcendental lattice TH . We skip the verification that T is primitive. It is similar to the proof ˜ of Lemma 4.1 from [Ko]. Thus the orthogonal complement of T in L is a primitive lattice of rank 16 with the same discriminant form as the Picard lattice SH . Now the result follows from the uniqueness theorem of Nikulin [Ni]. 7.2. Remark. Note that the orthogonal complement of the lattice R0 in L is isomorphic to the Picard lattice SK of a general Jacobian Kummer surface. It contains the sublattice SH as the orthogonal complement of the projection of r0 in SK . One can give an explicit formula for twenty vectors Nα , Tβ ∈ T ⊥ whose intersection matrix coincides with the intersection matrix of the divisor classes Nα , Tβ . We shall identify Nα , Tβ with Nα , Tβ . Explicitly, Nα , Tβ correspond to the Leech roots (2νK , 1, 1), where K ⊂ Ω is N45 : {∞, 0, 1, 3, 4, 11, 19, 20}, N24 : {∞, 0, 1, 3, 7, 9, 16, 18}, N56 : {∞, 0, 1, 3, 6, 8, 10, 13}, N26 : {∞, 0, 1, 3, 12, 15, 21, 22}, BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES N36 : {∞, 0, 1, 4, 5, 7, 8, 15}, N46 : {∞, 0, 1, 4, 10, 14, 18, 21}, N13 : {∞, 0, 1, 6, 11, 14, 15, 16}, T16 : {∞, 0, 2, 3, 4, 8, 9, 21}, 3047 N35 : {∞, 0, 1, 4, 6, 9, 12, 17}, N16 : {∞, 0, 1, 5, 6, 18, 20, 22}, N12 : {∞, 0, 1, 13, 15, 17, 18, 19}, T34 : {∞, 0, 2, 3, 6, 12, 16, 20}, T14 : {∞, 0, 2, 3, 7, 11, 13, 15}, T36 : {∞, 0, 2, 3, 10, 18, 19, 22}, T26 : {∞, 0, 2, 4, 5, 6, 10, 11}, T25 : {∞, 0, 2, 4, 7, 17, 18, 20}, T15 : {∞, 0, 2, 4, 12, 14, 15, 19}, T56 : {∞, 0, 2, 5, 15, 16, 18, 21}, T46 : {∞, 0, 2, 6, 8, 15, 17, 22}, T23 : {∞, 0, 2, 6, 9, 13, 14, 18}. For Leech roots r, r ∈ L corresponding to the Leech vectors λ, λ we have (r, r ) = 0, if λ − λ ∈ Λ4 ; 1, if λ − λ ∈ Λ6 . Here Λ4 = {x ∈ Λ : (x, x) = 4} = {(±28 , 016 ), (±3, ±123 ), (±42 , 022 )}, Λ6 = {x ∈ Λ : (x, x) = 6} = {(±2 , 0 ), (±33 , ±121 ), (±4, ±28 , 015 ), (±5, ±123 )}. 12 12 Let ω = (0, 0, 1) ∈ L. It is called the Weyl vector of the lattice L. It is an isotropic vector with the property that (ω, l) = 1 for any Leech root l. 7.3. Lemma. The projection ω of ω in SH is equal to the vector ˜ ∆=N +T. Proof. Note that (ω , Nα ) = (ω , Tβ ) = 1 for all α, β . On the other hand, the divisor N +T = Nα + Tβ has the same property. The result follows from Corollary 4.2. 3048 IGOR DOLGACHEV AND JONGHAE KEUM 8. Automorphisms of a general Hessian quartic surface Let X be a K3 surface with Picard lattice S . The automorphism group Aut(X ) of X has a natural representation ρ : Aut(X ) → O(S ) in the orthogonal group of S . Let W2 (S ) denote the subgroup generated by reflections in vectors r with r2 = −2. This group is a normal subgroup of O(X ), and the induced homomorphism ρ : Aut(X ) → O(S )/W2 (S ) has a finite kernel and a finite cokernel. This non-trivial result follows from the Global Torelli Theorem for algebraic K3 surfaces proven by I. Piatetski-Shapiro and I. Shafarevich [PS]. Let us describe the kernel and the cokernel. First of all, the group O(S )/W2 (S ) has the following interpretation. Let VS = {x ∈ S ⊗ R : + x2 > 0}, and let VS be its connected component containing an ample divisor class. + The group W2 (S ) has a fundamental domain P (S ) in VS (a cone over a convex + polytope in the corresponding Lobachevski space VS /R+ ). It can be chosen in such a way that its bounding hyperplanes are orthogonal to effective classes r with r2 = −2 and it contains the ray spanned by an ample divisor class. Let A(P (S )) ⊂ O(S ) be the group of symmetries of P (S ). Then O(S ) is equal to the semi-direct product W2 (S ) A(P (S )) of W2 (S ) and A(P (S )). The image of Aut(X ) in O(S ) is contained in A(P (S )). Let D(S ) = S ∗ /S be the discriminant group of the lattice, and qS the discriminant quadratic form on D(S ). We have a natural homomorphism A(P (S )) → O(qS ). Let Γ(S ) ⊂ A(P (S )) be the pre-image of {±1}. Then the image of Aut(X ) in A(P (S )) is contained in Γ(S ) as a subgroup of index ≤ 2. It is equal to the whole group Γ(S ) if Aut(X ) contains an element acting as −1 on D(S ). This will be the case for the Hessian, surface. The kernel of Aut(X ) → Γ(S ) is a finite cyclic group. It is trivial if X does not admit a nontrivial automorphism preserving any ample divisor. This happens in our case for a general Hessian, since it is known that a projective automorphism of a general cubic surface is trivial. Summing up, we obtain 8.1. Proposition. Let SH be the Picard lattice of a minimal nonsingular model ˜ ˜ H of a general Hessian quartic surface H . The group of automorphisms of H is isomorphic to the group Γ(SH ) of symmetries of P (SH ) which act as ±1 on the discriminant group of SH . Let WLee (L) be the subgroup of O(L) generated by reflections in Leech roots. + + Let P (L) be its fundamental domain in the VL (where VL is one of the components 2 of V = {x ∈ L ⊗ R : x > 0}) whose closure contains the Weyl vector ω . By a result of J. Conway, O(L) is equal to the semi-direct product WLee (L) A(P (L)) of WLee (L) and the group of symmetries A(P (L)) of P (L), and the latter is isomorphic to the group Λ O(Λ) of affine automorphisms of Λ. Now put P (SH ) = P (L) ∩ V (SH )+ . It is known that P (SH ) is non-empty, has only finitely many faces, and is of finite volume (in the Lobachevski space). Also it is known ([Bo], Lemmas 4.1-4.3) that the projection ω of the Weyl vector is contained in P (SH ) . Applying Lemma 8.3, we see that P (SH ) contains an ample divisor class, and hence P (SH ) is a part of P (SH ). BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3049 8.2. Lemma. Let G = Aut(P (SH ) ) be the group of symmetries of P (SH ) . Then G is isomorphic to Z/2 × S5 and can be realized as the subgroup of O(SH ) generated by the Enriques involution and the group of symmetries of the Sylvester pentahedron. It extends to a subgroup of O(L) which leaves the root lattice R invariant and induces the isometries of R defined by the symmetries of the Dynkin diagram of R. Proof. This is almost a word-for-word repetition of the proof of Lemma 4.5 in [Ko]. ˜ To find a certain set of generators of Aut(H ) containing the Enriques involution, we use the following strategy suggested in [Ko]. First we shall find the hyperplanes which bound P (SH ) . They correspond to rank 11 root sublattices R generated by + R and some Leech root r. The hyperplane {x ∈ VL : (x, r) = 0} is a boundary wall of P (L) and has non-empty intersection with P (SH ). Then for such a hyperplane ˜ we find an automorphism of H which maps one of the two half-spaces defined by this hyperplane to the opposite one. Let N be the group generated by these ˜ automorphisms. Then we check that for any automorphism γ ∈ Aut(H ) one can find δ ∈ N such that δ ◦ γ is a symmetry of P (SH ) . Applying Lemma 9.2, we conclude that δ ◦ γ is either the identity or the Enriques involution (since the latter and the identity are the only elements of Aut(P (SH ) ) which act as ±1 on the discriminant group of SH ). The next lemma is a simple repetition of the computations from [Ko], Lemma 4.6. 8.3. Lemma. Let r be a Leech root. Assume that r and R generate a root lattice R of rank 11. Then one of the following cases occurs: (0) (1a) (1b) (2) (3a) (3b) R R R R R R = A5 ⊕ A6 , where r is orthogonal to R; 1 = D6 ⊕ A5 , where (r, r0 ) = 0, (r, z ) = 1; 1 = D6 ⊕ A5 , where (r, r0 ) = 1, (r, z ) = 0; 1 = A3 ⊕ A3 ⊕ A5 , where (r, xi ) = (r, xj ) = 1, i, j = 0; 1 = A7 ⊕ A4 , where (r, x0 ) = (r, xi ) = 1; 1 = A7 ⊕ A4 , where (r, x) = (r, xi ) = 1. 1 Moreover, in case (0), r is one of the twenty Leech roots corresponding to Nα and Tβ . In case (1a), up to a symmetry of P (SH ) , we can choose r = (λ, 1, −1 − λ, λ ) 2 corresponding to the Leech vector λ = (ξ∞ , ξ0 , ξj1 , ξj2 , . . . , ξj6 , ξj7 , . . . , ξj22 ) = (3, 3, 3, −1, . . . , −1, 1, . . . , 1), where K = {∞, 0, j1 , . . . , j6 } is an octad satisfying |K ∩ K0 | = {∞, j1 }, |K ∩ Ki | = 4 and j1 ∈ Ki for i = 1, . . . , 5. In case (1b), r corresponds to Nα , α ∈ {0, 14, 15, 23, 25, 34}, or to Tβ , β ∈ {0, 12, 13, 24, 35, 45}. Together with “old ” Nα and Tβ they define 32 vectors spanning the Kummer overlattice SK of SH . 3050 IGOR DOLGACHEV AND JONGHAE KEUM In case (2), r corresponds to a Leech vector λ = 2νK , 0, ∞ ∈ K, |K ∩ Kl | = 4 |K ∩ Ki | = |K ∩ Kj | = 2, for l = 0, . . . , 5, l = i, j. In case (3a), if r meets x0 , xi , it corresponds to a Leech vector λ = νΩ − 4νk , k = 0, ∞, k ∈ Ki , k ∈ Kj , j = 0, 1, . . . , 5, j = i. In case (3b), if r meets x, xi , it corresponds to a Leech vector λ = 4ν0 − νK + νΩ−K , |K ∩ K0 | = 0, |K ∩ Ki | = 4, 0 ∈ K, ∞ ∈ K, |K ∩ Kl | = 2, l = 1, . . . , 5, l = i. 8.4. Remark. The number of vectors r in case (1a) is equal to 12. They correspond to the following octads K : {∞, 0, 1, 5, 9, 11, 13, 21}, {∞, 0, 1, 7, 10, 11, 17, 22}, {∞, 0, 1, 7, 12, 13, 14, 20}, {∞, 0, 1, 8, 9, 14, 19, 22}, {∞, 0, 1, 8, 16, 17, 20, 21}, {∞, 0, 1, 5, 10, 12, 16, 19}, {∞, 0, 2, 9, 11, 16, 17, 19}, {∞, 0, 2, 5, 8, 13, 19, 20}, {∞, 0, 2, 7, 8, 10, 14, 16}, {∞, 0, 2, 5, 7, 9, 12, 22}, {∞, 0, 2, 10, 12, 13, 17, 21}, {∞, 0, 2, 11, 14, 20, 21, 22}. The number of vectors r in case (2) is equal to 10 = corresponds to the following octad K : 5 2 . When i = 1, j = 2, r {0, ∞, 6, 7, 10, 12, 15, 18}. The number of vectors r in case (3a) is equal to 5 · 3 = 15. When i = 1, they correspond to the following Leech vectors: νΩ − 4νk , k ∈ {5, 14, 17}. The number of vectors r in case (3b) is equal to 5 · 3 = 15. When i = 1, they correspond to the following octads K : {0, 5, 9, 12, 13, 14, 17, 19}, {0, 5, 10, 11, 14, 16, 17, 21}, {0, 5, 7, 8, 14, 17, 20, 22}. BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3051 8.5. Lemma. Let r be a Leech root as in the previous lemma. Let r = r1 + r2 , ∗ where r1 ∈ SH and r2 ∈ T ∗ . Then Case (0) : r1 = r; Case (1a) : r1 = r + 1 (2x + 4z + 3y + 2r0 + x0 ), (r1 , r1 ) = −2/3; 3 Case (1b) : r1 = r + 1 (x + 2z + 3y + 4r0 + 2x0 ), (r1 , r1 ) = −2/3; 3 Case (2) : r1 = r + 1 (xi + xj ), (r1 , r1 ) = −1; 2 Case (3a) : r1 = r + 1 (x + 2z + 3y + 4r0 + 5x0 ) + 1 xi , (r1 , r1 ) = −2/3; 6 2 Case (3b) : r1 = r + 1 (5x + 4z + 3y + 2r0 + x0 ) + 1 xi , (r1 , r1 ) = −2/3. 6 2 Proof. Case (0) is obvious. ∗ Case (1a): Let x∗ , y ∗ , z ∗ , r0 , x∗ denote the dual basis of the basis x, y, z, r0 , x0 , i . . . , x5 . Since (r, z ) = 1, (r, x) = (r, y ) = (r, r0 ) = (r, xi ) = 0, we see that r2 = z ∗ = −(2x + 4z + 3y + 2r0 + x0 )/3 and hence (r1 , r1 ) = (r − r2 , r − r2 ) = (r, r) − (r2 , r2 ) = −2 + 4/3 = −2/3. Case (1b) Similar to the previous case. Case (2): Here r2 = x∗ + x∗ = −(xi + xj )/2. This gives (r1 , r1 ) = −1. i j Case (3a): r2 = x∗ + x∗ = − 1 (x + 2z + 3y + 4r0 + 5x0 ) − 1 xi . This gives 0 i 6 2 (r1 , r1 ) = −2/3. Case (3b) Similar to the previous case. Let sr : v → v + (v, r)r be the reflection of L in a Leech root r. Then the restriction of sr to SH ⊗ Q is a reflection sr1 (v ) = v + (v, r1 )r1 , where r1 is the projection of r onto SH ⊗ Q. This is, in general, not an isometry of SH . ˜ To find an automorphism of H corresponding to each vector in case (1a)-(3b), we need to express r1 in terms of Nα , Tβ . Case (1a). Let r correspond to the Leech vector 3ν0 + 3ν∞ + 3ν1 − ν5 − ν9 − ν11 − ν13 − ν21 + νΩ−K , where K = {0, 1, ∞, 5, 9, 11, 13, 21}. It is immediately verified that (r, Nα ) = 0, (r, Tβ ) = for all α, 1, if β = 16, 26, 56, 14, 23; 0, otherwise. This determines r1 in the form of a linear combination of Nα , Tβ . Writing down the corresponding system of linear equations and solving it, we obtain (8.1) r1 = 1 (−2T + 2N + 10(T36 + T46 + T15 + T25 + T34 ) 15 + 5(N36 + N46 + N56 + N13 + N24 )). Let α = 3r1 . 3052 IGOR DOLGACHEV AND JONGHAE KEUM By (3.1), N − T = 5ηH − 5ηS , so that α = 2ηH − 2ηS + 2(T36 + T46 + T15 + T25 + T34 ) + (N36 + N46 + N56 + N13 + N24 ). It is easy to check that α is a primitive vector of SH and a (−6)-root. Case (1b). These are conjugate to those in case (1a) by the Enriques involution τ , which can be viewed as an automorphism of P (SH ) . For example, take r = T12 . We know that (r1 , Tα ) = 0 for all α, and (r1 , Nβ ) = 1, if β = 16, 26, 12, 35, 45; 0, otherwise. Applying the Enriques involution τ , we see that (τ (r1 ), Nα ) = 0 for all α, and (τ (r1 ), Tβ ) = 1, if β = 16, 26, 56, 14, 23; 0, otherwise. Thus, τ (r1 ) is nothing but the vector that appeared in Case (1a) above. Case (2). Let r = (2νK , 1, 1), where K = {0, ∞, 6, 7, 10, 12, 15, 18}. We have (r, Nα ) = 1, if α = 45; 0, otherwise, (r, Tβ ) = 1, if β = 16; 0, otherwise. Notice that N45 is the vertex opposite to the edge T16 . It is easy to see that 1 1 (r, Nα ) = ( (C16 − N45 ), Nα ), (r, Tβ ) = ( (C16 − N45 ), Tβ ) 2 2 for all α and β . Thus 1 1 (8.2) r1 = (C16 − N45 ) = (ηH − 2T16 − N16 − N12 − N13 − N45 ). 2 2 Note that α = 2r1 is a (−4)-root of SH . Also note that τ (α) = α. Case (3a). Let r correspond to the Leech vector νΩ − 4ν5 . We have (r, Nα ) = 1, if α = 16, 36; 0, otherwise, (r, Tβ ) = 1, if β = 26, 56; 0, otherwise. By solving a system of linear equations we find that 1 1 r1 = (N26 + N45 + N56 + N24 ) − (N16 + N36 ) 2 3 (8.3) 2 1 1 + (N13 + N46 ) + (T46 + T25 ) + (T15 + T23 ). 3 3 3 It can be checked that the minimum positive integer k with kr1 ∈ SH is 6, so that 6r1 = 3(N26 + N45 + N56 + N24 ) − 2(N16 + N36 ) + 2(N13 + N46 ) + 4(T46 + T25 ) + 2(T15 + T23 ) BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3053 is a primitive (−24)-vector in SH . We remark that α = 6r1 is NOT a root. Case (3b). These are conjugate to those in case (3a) by the Enriques involution τ . For example, let r correspond to the Leech vector 4ν0 − νK + νΩ−K , where K = {0, 5, 9, 12, 13, 14, 17, 19}. We have (r, Nα ) = 1 if α = 12, 35; 0 otherwise, (r, Tβ ) = 1, if β = 15, 23; 0 otherwise. Applying τ , we see that (τ (r1 ), Tα ) = 1, if α = 26, 56; 0, otherwise, (τ (r1 ), Nβ ) = 1, if β = 16, 36; 0 otherwise. and that τ (r1 ) is the one that appeared in Case (3a) above. ˜ Let us find an involution σ of H corresponding to a hyperplane defined by a Leech root r of type (1a), (2), or (3a). If r1 denotes the projection of r to SH , we need that σ (r1 ) = −r1 . Automorphisms of type (1a). This is the inversion of an elliptic pencil of type 3. For example, the involution φ from (6.1) corresponds to the vector r1 from (8.1). Since α = 3r1 is a (−6)-root, we get φ(r1 ) = −r1 . Also (8.4) φ(ω ) = ω + 3(ω , r1 )r1 = ω + 15r1 , where ω = α Nα + β Tβ is the projection of the Weyl vector (see Lemma 8.3). Let us explain why the number of such automorphisms is 12. The inversion of the same elliptic fibration with respect to the different zero section T26 (and a 2torsion T56 ) gives another involution. Since there are 30 elliptic fibrations of type 3, we get in this way 60 involutions, but only 12 of them are different. Automorphisms of type (1b). This is the conjugate involution τ ◦ φ ◦ τ . Automorphisms of type (2). These are the 10 projections pα from a node Nα . For example, consider p45 , whose action can be computed as in Section 7-(ii), and take r1 computed in (8.2). It is easy to check that p45 (r1 ) = −r1 . Note that p45 ⊥ acts nontrivially on the hyperplane r1 in V (SH )+ . In other words, p45 is not a reflection, but works like a reflection. Also observe that (8.5) p45 (ω ) = ω + 4r1 = ω + 2(ω , r1 )r1 . 3054 IGOR DOLGACHEV AND JONGHAE KEUM Automorphisms of type (3a). This is the inversion of an elliptic pencil of type 2. To see this, take r1 computed in (8.3). Consider the elliptic fibration |FN12 ,T26 | = |FN35 ,T56 | = |T23 + N26 + T25 + N56 + T15 + N24 + T46 + N45 | = |C56 + T34 + N12 + T16 | = |C26 + T36 + N35 + T14 |. Observe that N13 , N46 , N16 , N36 are sections. Take N13 as the zero section. Then ˜ N46 is a 2-torsion. Let fr = fN16 ,N36 ,T26 ,T56 be the automorphism of the surface H defined by the inversion map with respect to the group law on the set of sections with zero section defined by N13 . Take D1 = (N26 + N45 ) + 2(N56 + N24 ) − N36 + (N13 + N46 ) + 2(T46 + T25 + T15 ), D3 = 2(N26 + N45 ) + (N56 + N24 ) − N16 + (N13 + N46 ) + 2(T46 + T25 + T23 ). Then D1 and D3 are effective (−2)-vectors, and D1 + N16 − 2N13 ∼ T36 − T14 + T23 + 2N26 + 3T25 + 2N56 + T15 + N24 + T46 + N45 , D3 + N36 − 2N13 ∼ T16 − T34 + T23 + 2N26 + 3T25 + 2N56 + T15 + N24 + T46 + N45 . We see that D1 and D3 are sections and D1 ⊕ N16 = D3 ⊕ N36 = 0, where ⊕ is the group operation on the set of sections with zero section defined by N13 . These determine the action of fr as follows: N16 ←→ D1 , N26 ←→ N56 , N36 ←→ D3 , N12 ←→ C56 , N13 ←→ N13 , N24 ←→ N45 , N46 ←→ N46 , N35 ←→ C26 , T15 ←→ T23 , fr (Tβ ) = Tβ , β = 16, 36, 46, 14, 25, 34, fr (T26 ) = 4ηH − 2(T26 + T56 ) − T26 − T15 − T23 − 2(N16 + N26 + N36 + N56 + N12 + N24 + N35 + N45 ), fr (T56 ) = 4ηH − 2(T26 + T56 ) − T56 − T15 − T23 − 2(N16 + N26 + N36 + N56 + N12 + N24 + N35 + N45 ). Note that D1 + D3 = N16 + N36 + 6r1 , so that fr (6r1 ) = −6r1 . Again fr is not a reflection, but works like a reflection. The involution fr is not symmetric in the sense that fr (T26 + T56 ) = T26 + T56 + 6r1 . On the other hand, the map gr = gN16 ,N36 ,T26 ,T56 := fN16 ,N36 ,T26 ,T56 ◦ p35 ◦ p12 is symmetric. We have N16 D3 N26 N45 N36 D1 N46 N13 N56 N24 N12 N12 N13 N46 N24 N56 N35 N35 N45 , N26 BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES T16 T34 T26 G2 T36 T14 T46 T25 T56 G5 T14 T36 T15 T15 T23 T23 T25 T46 3055 T34 , T16 where gr interchanges two elements in the same column, and G2 = T26 + T15 + T23 + 2(T46 + T25 ) + 2(N26 + N24 ) − (N16 + N36 ) + N56 + N45 + N13 + N46 , G5 = T56 + T15 + T23 + 2(T46 + T25 ) + 2(N56 + N45 ) − (N16 + N36 ) + N26 + N24 + N13 + N46 . This is an isometry of SH acting as −1 on the discriminant form of SH , and hence sending 6r1 to −6r1 . In fact, following an idea from [Ke2], we found first the lattice involution gr and then fr which realizes it geometrically. Since gr (N16 + N36 ) = N16 + N36 + 6r1 and gr (T26 + T56 ) = T26 + T56 + 6r1 , we have (8.6) gr (r1 ) = −r1 , where ω = α Nα + β gr (ω ) = ω + 12r1 = ω + 3(ω , r1 )r1 , Tβ is the projection of the Weyl vector. Automorphisms of type (3b). This is the involution τ ◦ gτ (r) ◦ τ. 8.6. Remark. The translation by the 2-torsion N46 is the same as τ ◦(involution defined by the skew lines T26 , T56 ) = p35 ◦ p12 . 8.7. Remark. If we take N16 as the zero section (and N36 a 2-torsion), then the inversion map corresponds to a Leech root of type (3b), more precisely, to r with (r, Nα ) = 1, if α = 46, 13; 0, otherwise, (r, Tβ ) = 1, if β = 26, 56; 0, otherwise. ˜ 8.8. Theorem. The automorphism group of H is generated by the Enriques involution τ , the 10 projections pα , the 15 inversion automorphisms fr of elliptic pencils of type 2, and the 12 inversion automorphisms φr of elliptic pencils of type 3. Proof. Let σr be the involution corresponding to a Leech root r of type (1a), (1b), (2), (3a), or (3b), i.e. σr is one of the twelve φr , twelve τ ◦ φτ (r) ◦ τ , ten pr , fifteen ˜ gr , or fifteen τ ◦ gτ (r) ◦ τ . Let N be the subgroup of Aut(H ) generated by them. As we explained before (after Lemma 8.2), the result follows from the following lemma. 8.9. Lemma. Let γ be an isometry of the Picard lattice SH which preserves P (SH ). Then there exists an element δ ∈ N such that δ ◦ γ ∈ Aut(P (SH ) ). Proof. This is similar to the proof of Lemma 7.3 in [Ko]. Take an element δ ∈ N which realizes min{(δ (γ (ω )), ω ) : δ ∈ N }. Then, for any r, (δ ◦ γ (ω ), ω ) ≤ (σr ◦ δ ◦ γ (ω ), ω ) = (δ ◦ γ (ω ), σr (ω )). If σr = φr (of type (1a)), then, applying (8.4), we get (δ ◦ γ (ω ), ω ) ≤ (δ ◦ γ (ω ), ω ) + 15(δ ◦ γ (ω ), r1 ). 3056 IGOR DOLGACHEV AND JONGHAE KEUM This means that (δ ◦ γ (ω ), r1 ) ≥ 0. Since ω is an interior point of P (SH ) , the last inequality is strict. If σr = τ ◦ φτ (r) ◦ τ (of type (1b)), then, since τ preserves ω , we have σr (ω ) = τ (ω + 15τ (r1 ) = ω + 15r1 , and hence again (δ ◦ γ (ω ), r1 ) > 0. The remaining cases can also be easily handled in this way by using (8.5) and (8.6). Now (δ ◦ γ (ω ), r1 ) > 0 for all r, so δ ◦ γ (ω ) ∈ P (SH ) . 8.10. Remark. As a general Hessian quartic surface degenerates to a general Jacobian Kummer surface, the involutions of type (1) are refined to become projections and correlations of the Jacobian Kummer, the involutions of type (2) become Cremona transformations related to G¨pel tetrads; on a Jacobian Kummer surface o there are 10 G¨pel tetrads having exactly 3 elements in common with the hexad o (the projection of the Leech root r0 from (7.1) onto the Picard lattice of the Jacobian Kummer surface is 1 (3ηK − 2hexad) and the projection of a Leech root of type 4 o (2) is 1 (ηK − G¨pel tetrad), and these two vectors are orthogonal to each other), 2 and the involutions of type (3a) also become Cremona transformations related to G¨pel tetrads; there are 15 G¨pel tetrads having exactly 2 elements in common with o o the hexad. It looks quite complicated to recover the involution fr or gr from its corresponding Cremona transformation on the Jacobian Kummer surface. Finally, the 192 new automorphisms [Ke2] on the Jacobian Kummer surface do not correspond to any automorphisms of the Hessian—in other words, as a general Hessian degenerates to a Jacobian Kummer surface, generators of type (3b) are replaced by new generators, which are the 192 automorphisms. Finally, note that in our case all the generators send one of the half-spaces defined by the corresponding hyperplane to the opposite half-space. The new automorphisms of the Jacobian Kummer do not act in this way; they send the half-space to a half-space corresponding to the inverse automorphism. References [Ba] [Bo] [Cos] [CD] [Hu1] [Hu2] [Ke1] [Ke2] [KK] [Ko] [Ni] [PS] H. Baker, Principles of Geometry. Vol III, Cambridge University Press, 1922, 2nd ed. 1954. MR 31:2650a R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133–153. MR 89b:20018 F. Cossec, Reye congruences, Trans. Amer. Math. Soc. 280 (1983), 737-751. MR 85b:14049 F. Cossec, I. 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Shafarevich, A Torelli theorem for algebraic surfaces of type K 3, Math. USSR Izv. 5 (1971), 547-587. MR 44:1666 BIRATIONAL AUTOMORPHISMS OF QUARTIC HESSIAN SURFACES 3057 J. Rosenberg, Hessian quartic surfaces which are Kummer surfaces, math. AG/9903037. G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, vol. 2, W. Metcalfe and Son, Cambridge, 5th ed., Longmans and Green, London, 1915; reprint, Chelsea, New York, 1965. MR 34:22 [To] J.A. Todd, A representation of the Mathieu group M24 as a collineation group, Ann. Mat. Pura Appl. (4) 71 (1966), 199-238. MR 34:2713 [vGe] B. van Geemen, private notes 1999. [Ro] [Sa] Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 E-mail address : [email protected] Korea Institute for Advanced Study, 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, Korea E-mail address : [email protected] ...
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