Unformatted text preview: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 354, Number 8, Pages 3031–3057
S 00029947(02)030118
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
secondorder 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 K3surface 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 nontrivial 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 K3surface
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 KRF2000041D00014.
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 halfspaces deﬁned by h to the other halfspace
−
deﬁned by h or to one of the two halfspaces 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 K3surfaces 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 K3surfaces 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
ﬁxedpointfree 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 trivalent 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 pullbacks 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 pullback 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 MordellWeil group MW(f ) of f is of rank
2 and is isomorphic to the MordellWeil 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 preimage 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
˜
preimage 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 preimage 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 preimages 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 preimage 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 preimage ∆ 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 preimage 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 preimage 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
2torsion points on its Jacobian variety J . In the usual way we identify the group of
2torsion points J2 with the set of 2element subsets α of the set S = {1, 2, 3, 4, 5, 6};
the zero point corresponds to the empty subset. We shall identify a 2element 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 dotproduct in F2 . The associated symmetric bilinear form is
2
( η , η )= ·η +η· . It is a nondegenerate 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 2torsion 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 AbelJacobi map corresponding to the point
p6 . The image of C is denoted by Θ0 . Let Θij = Θ0 + µij be its translate by
a 2torsion point µij . Each Θij contains exactly six 2torsion 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 rightside table contains a 2torsion point corresponding to the
entry (cd) in the leftside 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
lefthandside 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 righthandside 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 righthandside 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 lefthandside 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 lefthandside 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 thetadivisor 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 2torsion 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 K3surface birationally isomorphic to the Hessian quartic of a nonsingular cubic surface admits an Enriques involution, any K3surface
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 2torsion
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 preimage 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 2torsion 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 2torsion 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 2torsion 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 preimage of A1 splits in T16 + N26 , the preimage of A2 splits in T14 + N45 ,
the preimage of B1 splits in T36 + N56 , the preimage 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 nonskew 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 2torsion 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 ﬁveelement 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) CoxeterDynkin 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 nontrivial result follows from the
Global Torelli Theorem for algebraic K3 surfaces proven by I. PiatetskiShapiro
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 semidirect 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 preimage
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 semidirect 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 nonempty, has only ﬁnitely many faces, and is of
ﬁnite volume (in the Lobachevski space). Also it is known ([Bo], Lemmas 4.14.3)
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 wordforword 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 nonempty intersection with P (SH ). Then for such a hyperplane
˜
we ﬁnd an automorphism of H which maps one of the two halfspaces 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 2torsion. 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 2torsion 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 2torsion), 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 halfspaces deﬁned by the corresponding hyperplane
to the opposite halfspace. The new automorphisms of the Jacobian Kummer do
not act in this way; they send the halfspace to a halfspace corresponding to the
inverse automorphism.
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[Ro]
[Sa] Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email address : [email protected]
Korea Institute for Advanced Study, 20743 Cheongryangridong, Dongdaemungu,
Seoul 130012, Korea
Email address : [email protected] ...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of MichiganDearborn.
 Fall '04
 IgorDolgachev
 Math

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