The moduli space of (1,11)-polarized abelian surfaces is unirational

Mark Gross* and Sorin Popescu**

Mathematics Institute  Department of Mathematics
University of Warwick  Columbia University
Coventry, CV4 7AL, UK  New York, NY 10027
mgross@maths.warwick.ac.uk  psorin@math.columbia.edu

Abstract. We prove that the moduli space A_{11}^{lev} of (1, 11)-polarized abelian surfaces with level structure of canonical type is birational to Klein’s cubic hypersurface in P^{4}. Therefore, A_{11}^{lev} is unirational but not rational, and there are no Γ_{11}-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of A_{9}^{lev}.

Classical results of Tai, Freitag and Mumford and newer results of O’Grady, Gritsenko, Hulek and Sankaran say that moduli spaces of polarized abelian varieties are almost always of general type. However, for abelian varieties of small dimension and polarizations of small degree the situation is different and the corresponding moduli spaces usually have beautiful geometry.

In this paper we describe a projective model for the moduli of complex abelian surfaces with a polarization of type (1, 11), with level structure of canonical type. As a direct consequence we obtain the unirationality of this moduli space, which also turns out to be non-rational. However, unirationality already implies that there exist no Γ_{11}-cusp forms of weight 3.

Let A_{d} denote the moduli space of polarized abelian surfaces of type (1, d), and let A_{d}^{lev} be the moduli space of (1, d)-polarized abelian surfaces with canonical level structure. The map which forgets the level structure represents A_{d}^{lev} as a finite cover of A_{d}. Its general fiber is Γ_{d}/Γ_{d}^{lev} ≅ SL_{2}(Z_{d}), where Γ_{d} and Γ_{d}^{lev} denote the corresponding paramodular groups. In particular, if d is an odd prime number, then the forgetful morphism is a ramified cover of degree d(d^{2} − 1)/2. (See [LB], [Mum], [GP1] for the definition of canonical level structure, and basic results.)

* Supported by NSF grant DMS-9700761. ** Partially supported by NSF grant DMS-9610205 and MSRI, Berkeley.

Our main result is the following

Theorem 0.1 The moduli space A_{11}^{lev} is birational to Klein’s cubic hypersurface

K = V ( ∑_{i∈Z_{5}} x_{i}^{2}x_{i+1} = 0) ⊂ P^{4}.

In particular, A_{11}^{lev} is unirational but not rational.

The cubic hypersurface K ⊂ P^{4} was first studied by Klein [Kl] (see also [KlF], Band II) in connection with the z-embedding of the modular curve X(11) of level 11, which turns out to be defined by the 4 × 4-minors of the Hessian of the equation of K. In this respect, we note that K is the unique PSL_{2}(Z_{11})-invariant of degree three in P^{4}, and that furthermore PSL_{2}(Z_{11}) is its full automorphism group [Ad1]. The Klein cubic being smooth is unirational but not rational, cf. [CG], [Mur], [Bea].

Our result should be regarded in light of the following facts: A_{t} is not unirational (and in fact p_{g}(Ã_{t}) ≥ 1) if t ≥ 13 and t ≠ 14, 15, 16, 18, 20, 24, 30, 36 (Gritsenko [Gri1], [Gri2]), while Ã_{p}^{lev} is a 3-fold of general type for all primes numbers p ≥ 37 (Hulek-Sankaran [HS1], Gritsenko-Hulek, appendix to [Gri1]), where Ã_{p}^{lev} is a smooth projective model of a compactification of A_{p}^{lev}. See also [GH], and the survey paper [HS2] for related results, and [Bo] for a finiteness result in the same spirit. On the other hand, \overline{A_{5}^{lev}} ≅ P(H^{0}(F_{HM}(3))), where bar stands for the Igusa (=Voronoi) toroidal compactification and F_{HM} is the Horrocks-Mumford bundle on P^{4} ([HM], [HKW]), while A_{7}^{lev} is rational having as birational model a smooth V_{22}, a prime Fano 3-fold of index 1 and genus 12 which is rational (see [MS], [Schr] and [GP2] for details). It would be interesting to know how the Klein cubic “compares” with the toroidal compactification of A_{11}^{lev}.

In a series of forthcoming papers [GP2], [GP3], we will give details as to the structure of A_{d}^{lev}, 6 ≤ d ≤ 12, (excluding d = 9 and 11, which are covered here) and A_{d}, d = 14, 16, 18 and 20. In particular, we will prove their rationality or unirationality.

The moduli space of (1, 11)-polarized abelian surfaces is unirational 3

Finally, the methods used in this paper also provide an easy proof of the rationality (over Q(ξ) for ξ a primitive 9^{th} root of unity) of A_{9}^{lev}. The unirationality of this space also follows implicitly from O’Grady’s work [O’G]. He identifies A_{p^{2}}^{lev}, for p prime, with the moduli space A_{1}(p) of pairs of principally polarized abelian surfaces and rank two subspaces of the p-torsion points, non-isotropic for the Weil pairing. O’Grady studies the extension to (natural) toroidal compactifications of the finite natural forgetful map π from A_{1}(p) to the moduli space A_{1} of principally polarized abelian surfaces. Not all singularities of the toroidal compactification of A_{1}(p) are canonical, so O’Grady needs to describe carefully a partial desingularization all of whose singularities are canonical, before being able to apply Hurwitz’s formula for π to get an expression for the canonical class. For p = 3, our method in addition to being simpler also has the advantage of providing an explicit rational parametrization (over Q(ξ)).

Acknowledgments: We thank Igor Dolgachev, David Eisenbud, Klaus Hulek, and Kristian Ranestad for many useful discussions, and Allan Adler and Gregory Sankaran for a careful reading of a preliminary version of this paper. We are also grateful to Dave Bayer, Dan Grayson and Mike Stillman for Macaulay [BS], and Macaulay2 [GS] which helped us tremendously to understand the shape of the equations described in this paper. The second author also thanks the Mathematical Sciences Research Institute, Berkeley for its hospitality while part of this paper was being written.

1 Preliminaries

We review basic properties of polarized abelian surfaces, specializing to the case of polarizations of type (1, 11). For a more detailed review of this material as used in this paper, see [GP1], §1. In general, we use the notation and definitions of [LB] and [Mum].

Let (A, L) be a general abelian surface with a polarization of type (1, 11). Then |L| induces an embedding of A ⊂ P^{10} = P(H^{0}(L)^{∨}) as a projectively normal surface (cf. [Laz]) of degree 22 and sectional genus 12. (The projective normality of the general such abelian surface, follows also from the proof of [GP1], Theorem 6.5.) Riemann-Roch tells us that A is contained in 22 quadrics, which generate the homogeneous ideal of A ([GP1], Theorem 6.5).

The line bundle L induces a natural map from A to its dual, ϕ_{L} : A → Â, given by x ↦ t_{x}^{∗}L ⊗ L^{−1}, where t_{x} : A → A is the morphism given by translation by x ∈ A. Its kernel K(L) is isomorphic to Z_{11} × Z_{11}, and is dependent only on the polarization.

For every x ∈ K(L) there is an isomorphism t_{x}^{∗}L ≅ L. This induces a projective representation K(L) → PGL(H^{0}(L)), which lifts uniquely to a linear representation of K(L) after taking a central extension of K(L)

1  C^{∗}  G(L)  K(L)  0,

whose Schur commutator map is the Weil pairing. G(L) is the theta group of L and is isomorphic to the abstract Heisenberg group H̃(11), while the above representation is isomorphic to the Schrödinger representation of H̃(11) on V = C(Z_{11}), the vector space of complex-valued functions on Z_{11}. An isomorphism between G(L) and H̃(11), which restricts to the identity on centers induces a symplectic isomorphism between K(L) and Z_{11} × Z_{11}. Such an isomorphism is called a level structure of canonical type on (A, c_{1}(L)). (See [LB], Chapter 8, §3 or [GP1], §1.)

A decomposition K(L) = K_{1}(L) ⊕ K_{2}(L), with K_{1}(L) ≅ K_{2}(L) ≅ Z_{11} subgroups isotropic with respect to the Weil pairing, and a choice of a characteristic c ([LB], Chapter 3, §1) for L, define a unique basis {ϑ_{x}^{c} | x ∈ K_{1}(L)} of canonical theta functions for the space H^{0}(L) defined in [LB], Chapter 3, §2. This basis allows an identification of H^{0}(L) with V via ϑ_{γ}^{c} ↦ x_{γ}, where x_{γ} is the function on Z_{11} defined by x_{γ}(δ) = \{ \begin{matrix}1 & γ = δ \\ 0 & γ ≠ δ \end{matrix} for γ, δ ∈ Z_{11}. The x_{0}, …, x_{10} can also be identified with coordinates on P(H^{0}(L)^{∨}). Under this identification, the representation G(L) → GL(H^{0}(L)) coincides with the Schrödinger representation H̃(11) → GL(V). We will only consider the action of H_{11}, the finite subgroup of H̃(11) → GL(V) generated in the Schrödinger representation by σ and τ, where

σ(x_{i}) = x_{i−1}, τ(x_{i}) = ξ^{−i}x_{i},

for all i ∈ Z_{11} and ξ = e^{\frac{2πi}{11}} is a primitive root of unity of order 11. Notice that [σ, τ] = ξ, so H_{11} is a central extension

1  μ_{11}  H_{11}  Z_{11} × Z_{11}  0.

Thus the choice of a canonical level structure means that if A is embedded in P(H^{0}(L)^{∨}) using as coordinates x_{γ} = ϑ_{γ}^{c}, γ ∈ Z_{11}, then the image of A will be invariant under the action of the Heisenberg group H_{11} via the Schrödinger representation. (See [LB], Chapter 6, §7).

If moreover the line bundle L is chosen to be symmetric (and there are always finitely many choices of such an L for a given polarization type), then the embedding via |L| is also invariant under the involution ι, where

ι(x_{i}) = x_{−i}, i ∈ Z_{11},

which restricts to A as the involution x ↦ −x.

Let N(H_{11}) be the normalizer of H_{11} inside SL(V), where the inclusion H_{11} ⊂ SL(V) is the Schrödinger representation. An element α ∈ N(H_{11}) induces an outer automorphism of H_{11}, and hence an automorphism of Z_{11} × Z_{11} preserving the Weil pairing e^{D}, for D = (1, 11). The group of such automorphisms is SL_{2}(Z_{11}), and thus we get a map ψ : N(H_{11}) → SL_{2}(Z_{11}). As in [HM] §1, one sees that the kernel of this map is H_{11} and that ψ is surjective. This leads to extensions

[figure: 1; H_{11}; N(H_{11}); ψ; SL_{2}(Z_{11}); 1]

[figure: 0; Z_{11} × Z_{11}; N(H_{11})/μ_{11}; SL_{2}(Z_{11}); ‖; 1]

where in the bottom row N(H_{11})/μ_{11} is a semidirect product by the above (symplectic) action of SL_{2}(Z_{11}). Since H^{2}(SL_{2}(Z_{11}), C^{∗}) = 0 it follows that N(H_{11}) is in fact the semi-direct product H_{11} ⋊ SL_{2}(Z_{11}) (see [HM], §1 for details in the identical case of H_{5}).

Therefore the Schrödinger representation of H_{11} induces an 11 dimensional representation

ρ_{11} : SL_{2}(Z_{11}) SL(V).

In terms of generators and relations, cf. [BM], one has:

PSL_{2}(Z_{11}) = ⟨S, T | S^{11} = (S^{2}T S^{6}T)^{3} = 1, (ST)^{3} = T^{2} = 1⟩,

where

S = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )  and  T = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ,

while the representation ρ_{11} is given projectively by

ρ_{11}(S) = (ξ^{ij/2}δ_{ij})_{0≤i,j≤10}  and  ρ_{11}(T) = \frac{1}{\sqrt{11}}(ξ^{−ij})_{0≤i,j≤10},

where ξ is the above fixed 11^{th} root of unity. (See [Tan] and [Si] for details.)

The center of SL_{2}(Z_{11}) is generated by T^{2}, and ρ_{11}(T^{2}) = −ι. Thus the representation ρ_{11} is reducible. In fact, if V_{+} and V_{−} are the positive and negative eigenspaces, respectively, of the involution ι acting on V, then V_{+} and V_{−} are easily seen to be invariant under ρ_{11}, and moreover ρ_{11} splits as ρ_{+} ⊕ ρ_{−}, where ρ_{±} is the representation of SL_{2}(Z_{11}) acting on V_{±}. Note that ρ_{−} is trivial on the center of SL_{2}(Z_{11}), so it descends to give an irreducible representation

ρ_{−} : PSL_{2}(Z_{11}) GL(V_{−}).

For the reader’s convenience, we reproduce from the Atlas of finite groups [CNPW] the character table for PSL_{2}(Z_{11}):

Size of conjugacy class  1  55  110  132  132  110  60  60
Conjugacy class  I  γ_{1}  γ_{2}  γ_{3}  γ_{4}  γ_{5}  γ_{6}  γ_{7}

Character

χ_{1}  1  1  1  1  1  1  1  1
χ_{2}  5  1  -1  0  0  1  β  β̄
χ_{3}  5  1  -1  0  0  1  β̄  β
χ_{4}  10  -2  1  0  0  1  -1  -1
χ_{5}  10  2  1  0  0  -1  -1  -1
χ_{6}  11  -1  -1  1  1  -1  0  0
χ_{7}  12  0  0  α  α′  0  1  1
χ_{8}  12  0  0  α′  α  0  1  1

where α = \frac{1}{2}(−1 + \sqrt{5}), α′ = \frac{1}{2}(−1 − \sqrt{5}), and β = \frac{1}{2}(−1 + \sqrt{−11}). The conjugacy classes are represented by

\begin{matrix}I & γ_{1} & γ_{2} & γ_{3} & γ_{4} & γ_{5} & γ_{6} & γ_{7} \\ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) & ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) & ( \begin{matrix}1 & −1 \\ 1 & 0 \end{matrix} ) & ( \begin{matrix}3 & 0 \\ 0 & 4 \end{matrix} ) & ( \begin{matrix}5 & 0 \\ 0 & 9 \end{matrix} ) & ( \begin{matrix}3 & 2 \\ 4 & 3 \end{matrix} ) & ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) & ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) \end{matrix}

Similarly, the other direct summand

ρ_{+} : SL_{2}(Z_{11}) GL(V_{+})

is one of two mutually dual 6-dimensional irreducible representations of SL_{2}(Z_{11}). We refer the reader to [Dor] for the character table of SL_{2}(Z_{11}).

2 Moduli of (1, 11)-polarized abelian surfaces

From now on, let (A, L) be a general abelian surface with a polarization of type (1, 11) and with canonical level structure. As seen in the previous section, |L| embeds A ⊂ P^{10} = P(H^{0}(L)^{∨}) = P(V^{∨}) as a projectively normal surface of degree 22 and sectional genus 12, which is invariant under the action of the Heisenberg group H_{11} via the Schrödinger representation. In particular, H^{0}(I_{A}(n)) is also a representation of weight n of the Heisenberg group (i.e., a central element z ∈ C^{∗} acts by multiplication with z^{n}), and hence all its irreducible components will have dimension 11/ gcd(11, n). (See [LB], pg. 179, for this last fact.)

We will first determine equations for the locus of odd two-torsion points of (1, 11) polarized abelian surfaces. This is a set in P^{−} = P(V_{−}^{∨}). To analyze the equations which arise, we will need to make use of the SL_{2}(Z_{11}) symmetry present.

Notice that S^{2}(V) = H^{0}(O_{P(H^{0}(L)^{∨})}(2)) is 66-dimensional, and as a representation of weight 2, splits into six isomorphic 11 dimensional representations of H_{11}, each isomorphic to a twist V′ of the Schrödinger representation. On the other hand, by (2.2) of [HM], Hom_{H_{11}}(V′, V ⊗ V) is an N(H_{11})/H_{11} ≅ SL_{2}(Z_{11})-module and coincides with the representation ρ_{11} described in Section 1 Decomposing into the positive and negative eigenspaces of the involution ι we deduce that

V_{+} = Hom_{H_{11}}(V′, S^{2}(V)) and V_{−} = Hom_{H_{11}}(V′, ∧^{2}(V)),

as SL_{2}(Z_{11})-modules. By (2.2) of [HM], φ : V′ ⊗ Hom_{H_{11}}(V′, S^{2}(V)) → S^{2}(V) is an isomorphism of N(H_{11})-modules, and since S^{2}(V) is an irreducible N(H_{11})-module it follows that V_{+} is a six dimensional irreducible representation of SL_{2}(Z_{11}).

The SL_{2}(Z_{11})-isomorphism φ : V_{+} ⊗ V′ → S^{2}(V) can be represented as follows. We use the usual basis x_{0}, …, x_{10} for V (identified with the basis of canonical theta functions of H^{0}(L)), and the basis f_{0}, …, f_{10} of V′ such that σ(f_{i}) = f_{i−1} and τ(f_{i}) = ξ^{−2i}f_{i}, i ∈ Z_{11}. Then there is a basis e_{0}, …, e_{5} of V_{+} (in fact the projection of x_{0}, …, x_{5} onto V_{+} coming from the decomposition V = V_{+} ⊕ V_{−}) such that the map φ takes e_{i} ⊗ f_{j} to the (i, j)^{th} entry of the 6 × 11 matrix R_{5}, whose entries are

(R_{5})_{ij} = x_{j+i}x_{j−i}, 0 ≤ i ≤ 5, 0 ≤ j ≤ 10,

where the indices of the variables are mod 11. Thus the span of the entries in any row of R_{5} are a H_{11}−subrepresentation of S^{2}(V). Also, any H_{11}−subrepresentation of S^{2}(V) can be obtained by taking a linear combination of the rows, and taking the span in S^{2}(V) of the resulting 11 quadratic polynomials. In addition, if P ∈ P^{10} and v ∈ V_{+}, then v ·R_{5}(P) = 0 if and only if P is contained in the scheme cut out by the H_{11}-subrepresentation of quadrics determined by v.

Abusing notation, we will also denote by V_{±} the eigenspaces of the involution ι acting on V′ (by ι(f_{i}) = f_{−i}). Then the restriction of φ to V_{+} ⊗ V_{+} induces a SL_{2}(Z_{11})- isomorphism

Φ : ∧^{2}(V_{+}) S^{2}(V_{−}),

usually called the intertwining operator (see [We], [AR] pp. 62–63, 74, or [Ad5] for details). We may regard the intertwining isomorphism Φ as being induced by a skew-symmetric matrix with entries quadratic polynomials in the coordinates of V_{−}, namely: Φ takes the element e_{i} ∧ e_{j} of ∧^{2}(V_{+}) to the (i, j)^{th} entry of the matrix

S = ( \begin{matrix}0 & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} & x_{5}^{2} \\ −x_{1}^{2} & 0 & x_{1}x_{3} & x_{2}x_{4} & x_{3}x_{5} & −x_{4}x_{5} \\ −x_{2}^{2} & −x_{1}x_{3} & 0 & x_{1}x_{5} & −x_{2}x_{5} & −x_{3}x_{4} \\ −x_{3}^{2} & −x_{2}x_{4} & −x_{1}x_{5} & 0 & −x_{1}x_{4} & −x_{2}x_{3} \\ −x_{4}^{2} & −x_{3}x_{5} & x_{2}x_{5} & x_{1}x_{4} & 0 & −x_{1}x_{2} \\ −x_{5}^{2} & x_{4}x_{5} & x_{3}x_{4} & x_{2}x_{3} & x_{1}x_{2} & 0 \end{matrix} ) .

Here we are abusing notation by identifying x_{i} ∈ V with the projection of x_{i} onto V_{−} under the decomposition V = V_{+} ⊕ V_{−}. Note that S can also be viewed as the restriction of the first 6 × 6-block of R_{5} to P^{−}, where we use x_{1}, …, x_{5} as coordinates on P^{−} (where x_{i} = x_{−i}).

We will only need the fact that S arises from the intertwining operator in Lemma 2.1 1) below. The key fact we will need later about S is the simple observation that if P ∈ P^{−}, then v · R_{5}(P) = 0 if and only if v · S(P) = 0.

Following [GP1], §6, we define D_{i} ⊆ P^{−} to be the locus where the matrix S has rank ≤ 2i, for i = 1, 2. By the previous observation, we may interpret D_{i} as the locus of points in P^{−} ⊂ P^{10} = P(V^{∨}) which are contained in at least a (6 − 2i)-dimensional family of H_{11}-representations of quadrics. Remark also that all the loci D_{i} are invariant under the action of PSL_{2}(Z_{11}) via the representation ρ_{−} defined in Section 1

Lemma 2.1

1) D_{1} ⊂ P^{−} is a smooth curve of degree 20 and genus 26 isomorphic to the modular curve X(11). (This is Klein’s z-model of the modular curve X(11). It is the “trace” of the origins in the Shioda compactification of elliptic normal curves with level structure in P^{10}, and its embedding is induced by λ^{4}, where λ, a 10^{th} root of the canonical bundle on X(11), is the generator of the group of PSL_{2}(Z_{11})-invariant line bundles on X(11). See [Kl], pp. 153-156, [AR] and [Dol] for details).

2) D_{2} ⊂ P^{−} is an irreducible sextic hypersurface, defined by the 6 × 6 Pfaffian of S. It contains as an open subset the locus in P^{10} of odd 2-torsion points of (1, 11)-polarized abelian surfaces with canonical level structure.

Proof: 1) Consider the composition

Ψ : P^{−} = P(V_{−}^{∨}) P(S^{2}(V_{−}^{∨})) ^{≅} P(∧^{2}(V_{+}^{∨})),

where the first map is the Veronese embedding and the second isomorphism is induced by the intertwining operator. Then for P ∈ P^{−}, one can write Ψ(P) = S(P), the latter being a skew-symmetric matrix which should be interpreted as an element of ∧^{2}(V_{+}^{∨}). It is then clear that D_{1} is the pull-back under Ψ of the locus in P(∧^{2}(V_{+}^{∨})) of rank 2 skew-symmetric matrices. This latter locus can be identified with Gr(2, V_{+}^{∨}). Thus D_{2} is isomorphic to the pull-back of Gr(2, V_{+}^{∨}) under Ψ.

Adler-Ramanan [AR], Theorem 19.17, study this pull-back. In particular, they show that this pull-back gives the same variety in P^{−} as that defined by Klein’s equations studied by Vélu in [Ve]. It is proven in [Ve], (summarized in Théorème 10.6), that Klein’s equations give the so-called the z-model of X(11), and this model is always nonsingular. (See [Dol], Theorem 5.1 for the explicit statement.) Thus D_{1} is isomorphic to X(11).

The genus of X(11) is well-known: see for example [Hu], pg. 59, for the genus of the modular curve of level n. The degree of the z-model is calculated in [Ve]. (See also [AR], Corollary 23.28.)

2) For a general point P ∈ P^{−} = P(V_{−}), S has rank 6, the intertwining operator being an isomorphism, thus P^{−} = D_{3} ≠ D_{2}. In fact D_{2} ⊂ P^{−} is the sextic hypersurface given by the Pfaffian of S. For the record, the equation f_{6} of D_{2} is

f_{6} = −x_{1}^{2}x_{2}x_{3}^{3} + x_{1}^{3}x_{3}x_{4}^{2} − x_{2}^{3}x_{3}^{2}x_{5} + x_{1}x_{4}^{3}x_{5}^{2} + x_{2}^{2}x_{4}x_{5}^{3}
+ x_{1}x_{2}^{4}x_{4} − x_{2}x_{3}x_{4}^{4} − x_{1}^{4}x_{2}x_{5} + x_{3}^{4}x_{4}x_{5} + x_{1}x_{3}x_{5}^{4}
+ x_{1}x_{2}x_{3}^{2}x_{4}^{2} − x_{1}^{2}x_{2}^{2}x_{3}x_{5} − x_{1}x_{2}^{2}x_{4}^{2}x_{5} − x_{1}^{2}x_{3}x_{4}x_{5}^{2} + x_{2}x_{3}^{2}x_{4}x_{5}^{2},

though we will not make use of its explicit form. To prove its irreducibility we will use the fact that it is a PSL_{2}(Z_{11})-invariant in S^{6}(V_{−}). Without loss of generality we may assume that as a PSL_{2}(Z_{11}) representation V_{−} has character χ_{3}. It is easy to see that χ_{S^{2}(V_{−})} = χ_{3} + χ_{5}, so in particular there are no PSL_{2}(Z_{11}) invariants in S^{2}(V_{−}). On the other hand, χ_{S^{3}(V_{−})} = χ_{1} + χ_{5} + χ_{6} + χ_{7}, so in S^{3}(V_{−}) there is precisely one PSL_{2}(Z_{11})- invariant, which we will denote by f_{3}. Thus the only way that f_{6} could fail to be irreducible is if f_{6} = f_{3}^{2}. But f_{6} is not a square. To see this, set, say x_{4} = x_{5} = 0 in the matrix S and take its Pfaffian, or just set x_{4} = x_{5} = 0 in the above equation for f_{6}. We get only one term, −x_{1}^{2}x_{2}x_{3}^{3}, which is not a square, so f_{6} itself cannot be a square, and thus f_{6} is irreducible. (Alternatively, the irreducibility of D_{2} follows from [Ad3].)

For any point P ∈ D_{2} \ D_{1}, S(P) is of rank 4, thus P is contained in a pencil of H_{11}-subrepresentations of quadrics in P^{10}. The last claim follows now from the fact that H_{11} ⋊⟨ι⟩-invariant abelian surfaces in P^{10} lie on a pencil of H_{11}-representations of quadrics, and that odd 2-torsion points of a general abelian surface get mapped to D_{2} \ D_{1} ⊂ P^{−}, and uniquely determine the surface (cf. [GP1], Lemma 6.3 and Lemma 6.4).

By Lemma 2.1, we may define now the morphism

Θ : D_{2} \ D_{1} Gr(2, V_{+}) = Gr(2, 6)
D_{2} \ D_{1} ∋ P ker(S(P)) = {v ∈ V_{+} | v · S(P) = 0},

which sends a point P to the pencil of H_{11}-subrepresentations of quadrics containing it. The general (1, 11)-polarized abelian surface embedded with level structure via a symmetric line bundle meets P^{−} in the (images of the) 6 odd 2-torsion points. By [GP1], Lemma 6.4 they are mapped via Θ to a single point in Gr(2, 6). Thus Θ factorizes as a rational map

Θ_{11} : A_{11}^{lev} Gr(2, 6) = Gr(2, V_{+}),

which essentially takes an abelian surface A to the point in Gr(2, 6) corresponding to the H_{11}-subrepresentation H^{0}(I_{A}(2)) ⊂ H^{0}(O_{P(V)}(2)).

Theorem 2.2 The map Θ_{11} : A_{11}^{lev} Gr(2, 6) yields a birational map between A_{11}^{lev} and im(Θ).

Proof: This is an immediate consequence of the degeneration arguments in [GP1], Theorem 6.5, and the above results.

In order to prove Theorem 0.1 we will need to determine the precise structure of im(Θ). To do so, we will use the following representation of the Plücker embedding of the Grassmannian Gr(2, 2m) = Gr(2, W), where W ≅ C^{2m}, m ≥ 2.

Gr(2, 2m) is embedded in P^{\binom{2m}{2}−1} = P(∧^{2}(W)) via the Plücker embedding, as the variety of those 2-vectors which are totally decomposable. Thus hyperplanes in the Plücker embedding can be identified with (projectivized) skew symmetric forms H ∈ P(∧^{2}(V)^{∗}), and thus with 2m × 2m skew-symmetric matrices. In this setting the Grassmannian Gr(2, 2m) can be also identified (as an embedded variety) with the subvariety R_{1} of 2m×2m skew-symmetric matrices of rank two. The hyperplane sections corresponding to points of R_{1} are the Schubert cycles σ_{1}, the sets of lines intersecting a given subspace of codimension 2 in P(W).

More precisely, if P ∈ Gr(2, 2m) corresponds to a subspace of W ≅ C^{2m} spanned by the rows of a 2 × 2m matrix

L = ( \begin{matrix}a_{1} & ⋯ & a_{2m} \\ b_{1} & ⋯ & b_{2m} \end{matrix} ) ,

then the corresponding 2m × 2m skew-symmetric matrix H_{P} = (p_{ij}) has as entries the Plücker coordinates of L

p_{ij} = a_{i}b_{j} − a_{j}b_{i}.

This matrix is rank 2, all the rows being linear combinations of the rows of L. Conversely, given any 2m × 2m skew-symmetric matrix of rank 2, the span of the rows yields a twodimensional subspace of C^{2m} = W, and hence a point in Gr(2, 2m). Furthermore, with obvious abuse of notation, the following correspondence holds:

Lemma 2.3 For H ∈ P(∧^{2}(W)^{∗}) and k ∈ {1, …, m} the following are equivalent

1) H ∈ R_{k} := {H ∈ P(∧^{2}(W)^{∗}) | rank(H) ≤ 2k}

2) H lies in the k-chordal locus of R_{1} ≅ Gr(2, 2m) (i.e., lies in a linear subspace P^{k−1} ⊂ P(∧^{2}(W)^{∗}) which is k-secant to R_{1}).

Proof: All these facts are classical, and easy to prove. See for instance [SR].

An easy computation shows that the codimension of R_{m−k} in P(∧^{2}(W)^{∗}) is \binom{2k}{2}. In particular, R_{m−1} is a hypersurface in P(∧^{2}(W)^{∗}) of degree m, defined by the Pfaffian of the generic skew-symmetric matrix.

Remark 2.4 The Plücker embedding is compatible with the natural action of PGL(W), and the orbits under this action are exactly R_{k} \ R_{k−1}, for k = \overline{1, m}, where R_{0} = ∅.

The following lemma describes the map Θ in the above setting:

Lemma 2.5 Let M be a 2m × 2m skew-symmetric matrix of forms of rank 2m − 2 on a variety X. Then the map Θ : X → Gr(2, 2m) induced by x ∈ X ↦ ker M(x) ⊆ C^{2m} is given in (dual) Plücker coordinates by

x M^{∗}(x) ∈ R_{1} = Gr(2, 2m),

where M^{∗} is the 2m × 2m skew-symmetric matrix defined by

M_{ij}^{∗} = \{ \begin{matrix}(−1)^{i+j}Pf^{ij}(M) & i < j, \\ 0 & i = j, \\ (−1)^{i+j+1}Pf^{ij}(M) & i > j, \end{matrix}

and where Pf^{ij}(M) is the Pfaffian of the matrix obtained by deleting the i^{th} and j^{th} rows and columns from M.

Proof: We’ll make use of the following standard facts concerning Pfaffian identities (see [BE] and [Re] for more details). Let F be a free module of rank n = 2m over the ring R, let F^{∗} denote the dual module, and let f : F^{∗} → F be a skew-symmetric morphism (that is the matrix of f corresponding to the choice of a basis in F and the dual basis in F^{∗} is a skew-symmetric n × n-matrix M). Now, ∧F^{∗} and ∧F are modules over each other and we adopt here [BE]’s notation in writing a(b) for the result of an operation of a ∈ ∧F on b ∈ ∧F^{∗} and vice-versa; thus a(b) ∈ ∧F^{∗} and b(a) ∈ ∧F. The skew-symmetric map f corresponds to an element φ ∈ ∧^{2}F, such that for all a^{∗} ∈ F^{∗} we have f(a^{∗}) = −a^{∗}(φ). In terms of a basis e_{1}, …, e_{2m} of F, if (f_{ij}) is the matrix of f with f_{ij} = −f_{ji}, then

φ = ∑_{i<j} f_{ij}e_{i} ∧ e_{j}.

Now fix an orientation e^{∗} ∈ ∧^{2m}F^{∗} (that is a generator of this module). This yields a correspondence between skew-symmetric maps g : F → F^{∗} and elements ψ ∈ ⋀^{2m−2} F, via g(a) = ψ(a(e^{∗})). In coordinates, write

ψ = ∑_{i<j} g_{ij}e_{{i,j}^{∗}},

where {i, j}^{∗} denotes the complement of the set {i, j} in {1, …, 2m}, and for a subset H = {i_{1}, …, i_{n}} ⊆ {1, …, 2m} with i_{1} < ⋯ < i_{n}, e_{H} = e_{i_{1}} ∧ ⋯ ∧ e_{i_{n}}. Notice that the matrix corresponding to g is ((−1)^{i+j}g_{ij}). We then have, for a ∈ F,

\begin{aligned}(f ∘ g)(a) & = f(ψ(a(e^{∗}))) \\ & = −ψ(a(e^{∗}))(φ). \end{aligned}

Recall that divided powers are related to Pfaffians by the formula

φ^{(p)} = ∑_{|H|=2p} Pf (M_{H})e_{H},

where M denotes the skew-symmetric matrix of f with respect to {e_{i}} and the dual basis {e_{i}^{∗}}, while M_{H} denotes the principal submatrix of M determined by rows and columns indexed by H. Thus, if M has rank 2m − 2, then φ^{(m)} = 0. Now Lemma 2.4 of [BE] tells us that for a ∈ F,

φ^{(m)}(a(e^{∗})) = φ^{(m−1)}(a(e^{∗}))(φ),

so if g : F → F^{∗} is the morphism corresponding to φ^{(m−1)} and φ^{(m)} = 0, then f ∘ g = 0.

In coordinates, let M be the skew symmetric n × n-matrix corresponding to f (and the above choice of bases), denote by Pf (M) its Pfaffian and by Pf^{ij}(M), for i < j, the Pfaffian of the skew-symmetric matrix obtained from M by ommiting the i^{th} and j^{th} rows and columns, and set Pf^{ij}(M) = −Pf^{ji}(M), if i > j.

Thus if the matrix M^{∗} = (m_{ij}^{∗}) is the skew-symmetric matrix with entries

m_{ij}^{∗} = \{ \begin{matrix}(−1)^{i+j}Pf^{ij}(M) & if i ≠ j, \\ 0 & if i = j, \end{matrix}

then M^{∗} is the matrix of g associated with the bases {e_{i}} of F and {e_{i}^{∗}} of F^{∗}. The above compositions read as: M · M^{∗} = 0. Thus also M^{∗} · M = (M · M^{∗})^{t} = 0, as required.

We show next that \overline{im(Θ)} is a smooth Fano 3-fold of genus 8 and index one. As a corollary of Theorem 2.2 and Theorem 2.6 below we obtain then Theorem 0.1.

Theorem 2.6 The Zariski closure of im(Θ) ⊂ Gr(2, 6) = Gr(2, V_{+}) in the Plücker embedding has equations given in Plücker coordinates by

p_{23} = −p_{15},  p_{26} = p_{13},  p_{14} = −p_{35},  p_{16} = p_{45},  p_{46} = −p_{12}.

Furthermore, this linear section of Gr(2, 6) is three-dimensional, smooth, and hence a Fano 3-fold of type V_{14}. Furthermore it is birational to the Klein cubic hypersurface

K = V ( ∑_{i∈Z_{5}} x_{i}^{2}x_{i+1} = 0) ⊂ P^{4}.

Proof: The first thing to check is that im(Θ) ⊂ P^{9} satisfies the five linear relations given above. This can easily be checked by hand using Lemma 2.5 simply by computing the corresponding Pfaffians of the matrix S and showing they satisfy the given relations. Observe also that the SL_{2}(Z_{11}) representation on ∧^{2}(V_{+}) induced by ρ_{+} decomposes as the sum of a 5 and a 10-dimensional irreducible representations. The equations in the statement of Theorem 2.6 define this 10-dimensional representation as a subspace of ∧^{2}(V_{+}).

To conclude that the closure of the image of Θ is actually given by these equations, we will show that the subscheme X of Gr(2, 6) defined by these equations is three-dimensional and non-singular, and thus a Fano 3-fold V_{14} of genus 8 and index 1. To this end we will use a classical construction due to G. Fano [Fa], and recast in modern language by Iskovskih [Is1], [Is2] (see also [Pu]), which shows that any V_{14} is birationally equivalent to a (smooth) cubic threefold in P^{4}.

Let H_{1}, …, H_{5} be five (linearly independent) hyperplanes in P^{14} = P(H^{0}(O_{Gr(2,6)}(1))), and let

X := Gr(2, 6) ∩ H_{1} ∩ ⋯ ∩ H_{5}.

By Lemma 2.3 and the discussion preceding it, we may identify (P^{14})^{∨} with the space of 6 × 6 skew-symmetric matrices, and so Gr(2, 6)^{∨} can be naturally identified with the locus R_{2} of skew-symmetric matrices of rank ≤ 4. As seen above Gr(2, 6)^{∨} is then a cubic hypersurface in (P^{14})^{∨} defined by the 6 × 6 Pfaffian of the generic skew-symmetric matrix. Its singular locus is the locus R_{1} of 6 × 6 skew-symmetric matrices of rank ≤ 2, which is isomorphic (as an embedded variety) to Gr(2, 6). By Lemma 2.3, the Pfaffian cubic is also the secant variety to R_{1} ≅ Gr(2, 6).

Let P^{4} := ⟨H_{1}, …, H_{5}⟩ ⊆ (P^{14})^{∨} denote the span of the above five hyperplanes as points in (P^{14})^{∨}, and let

B := Gr(2, 6)^{∨} ∩ ⟨H_{1}, …, H_{5}⟩.

Define now a (possibly rational) map

Ψ : B Gr(2, 6)
B ∋ b ker(b) ∈ Gr(2, 6),

where we think of each element b of B as a 6 × 6 skew-symmetric matrix of rank ≤ 4. Therefore Ψ is defined on all of B iff B is disjoint from the singular locus of Gr(2, 6)^{∨}.

Lemma 2.7 If the cubic hypersurface B is smooth and im(Ψ) ∩ X = ∅, then X is a non-singular threefold (Fano of genus 8, index 1).

Proof: First, let x ∈ X be a point where the Zariski tangent space of X at x, T_{X,x}, has dim T_{X,x} > 3. If l_{1}, …, l_{5} are the Zariski tangent spaces to H_{1}, …, H_{5} at x, then

T_{X,x} = T_{Gr(2,6),x} ∩ l_{1} ∩ … ∩ l_{5} ⊆ T_{P^{14},x}.

Suppose now that H_{1}, …, H_{5} are given by the linear equations h_{1} = 0, …, h_{5} = 0, respectively. The only way T_{X,x} could fail to be three dimensional is if there exist a hyperplane section H whose equation is ∑_{i=1}^{5} a_{i}h_{i} = 0 for some a_{i}, with T_{H,x} ⊇ T_{Gr(2,6),x}. Thus H must be tangent to Gr(2, 6), and so H ∈ B. If the cubic B is smooth, then B is disjoint from the singular locus of Gr(2, 6)^{∨}, and the map Ψ above is defined everywhere on B. Now Ψ(H) is the point of the Grassmannian Gr(2, 6) which H is tangent to; thus, in particular, if H is tangent to Gr(2, 6) at a point of X, we must have Ψ(H) ∈ X and Ψ(B) ∩ X ≠ ∅. In conclusion, if Ψ(B) ∩ X = ∅, we must have dim T_{X,x} = 3 and so X is a non-singular Fano threefold.

Proof of Theorem 2.6 continued: We use Lemma 2.7 to check that the subscheme X of Gr(2, 6) defined by the equations in the statement of Theorem 2.6 is three-dimensional and non-singular. We can now compute B directly in our case: Let x_{ij}, 1 ≤ i < j ≤ 6, be coordinates on (P^{14})^{∨} dual to the Plücker coordinates p_{ij} on P^{14}. Then, in our particular case, the P^{4} spanned by H_{1}, …, H_{5} is cut out by the equations

\begin{aligned}x_{12} − x_{46} & = 0, \\ x_{13} + x_{26} & = 0, \\ x_{14} − x_{35} & = 0, \\ x_{15} − x_{23} & = 0, \\ x_{16} + x_{45} & = 0, \\ x_{24} = x_{25} & = x_{34} = x_{36} = x_{56} = 0. \end{aligned}

Making now the substitutions x_{0} = x_{12}, x_{2} = x_{13}, x_{1} = x_{14}, x_{4} = x_{15}, and x_{3} = x_{16}, we see that the equation of B ⊆ P^{4}, with coordinates x_{0}, …, x_{4}, is given by the 6 × 6 Pfaffian of the skew-symmetric matrix

M = ( \begin{matrix}0 & x_{0} & x_{2} & x_{1} & x_{4} & x_{3} \\ −x_{0} & 0 & x_{4} & 0 & 0 & −x_{2} \\ −x_{2} & −x_{4} & 0 & 0 & x_{1} & 0 \\ −x_{1} & 0 & 0 & 0 & −x_{3} & x_{0} \\ −x_{4} & 0 & −x_{1} & x_{3} & 0 & 0 \\ −x_{3} & x_{2} & 0 & −x_{0} & 0 & 0 \end{matrix} )

which is

B = {x_{0}^{2}x_{1} + x_{1}^{2}x_{2} + x_{2}^{2}x_{3} + x_{3}^{2}x_{4} + x_{4}^{2}x_{0} = 0}.

Thus B is Klein’s cubic K = {∑_{i=0}^{4} x_{i}^{2}x_{i+1} = 0}, the only P SL_{2}(Z_{11})-invariant cubic in P^{4}. This cubic is known to be smooth. (See also [Ad4], Lemma 47.2 for the Pfaffian description of the Klein cubic.)

To show that X is non-singular, we need now to check the second hypothesis of Lemma 2.7. By Lemma 2.5, the map Ψ : B → Gr(2, 6) is given in Plücker coordinates by the matrix

(M)^{∗} = ( \begin{matrix}0 & x_{0}x_{1} & x_{2}x_{3} & x_{1}x_{2} & x_{0}x_{4} & x_{3}x_{4} \\ −x_{0}x_{1} & 0 & x_{3}^{2} + x_{0}x_{4} & x_{1}x_{3} & −x_{0}x_{2} & −x_{1}^{2} − x_{2}x_{3} \\ −x_{2}x_{3} & −x_{3}^{2} − x_{0}x_{4} & 0 & −x_{2}x_{4} & x_{0}^{2} + x_{1}x_{2} & x_{0}x_{3} \\ −x_{1}x_{2} & −x_{1}x_{3} & x_{2}x_{4} & 0 & −x_{2}^{2} − x_{3}x_{4} & x_{0}x_{1} + x_{4}^{2} \\ −x_{0}x_{4} & x_{0}x_{2} & −x_{0}^{2} − x_{1}x_{2} & x_{2}^{2} + x_{3}x_{4} & 0 & −x_{1}x_{4} \\ −x_{3}x_{4} & x_{1}^{2} + x_{2}x_{3} & −x_{0}x_{3} & −x_{0}x_{1} − x_{4}^{2} & x_{1}x_{4} & 0 \end{matrix} ) .

In order for a point P = (x_{0} : …: x_{4}) ∈ B to satisfy Ψ(B) ∈ X, the Plücker coordinates of Ψ(P) must satisfy the five linear equations defining X, which yields that P must satisfy the equations

\begin{aligned}x_{i}^{2} + 2x_{i+1}x_{i+2} & = 0, 0 ≤ i ≤ 4 \\ ∑_{i=0}^{4} x_{i}^{2}x_{i+1} & = 0 \end{aligned} .

The first set of equations is precisely the Jacobian of the Klein cubic B, and since B is smooth, there are no points P ∈ B satisfying these equations. Hence, by Lemma 2.7, X is non-singular.

We are left now to construct a birational map between B(= K) ⊂ P^{4} and X ⊂ P^{9}. As mentioned above, a classical construction due to Fano and Iskovskih [Fa], [Is2] (see also [Pu] for details) provides such a (rather indirect) birational transformation. For the reader’s convenience we sketch it in the sequel.

The lines L_{p} in P^{5} = P(V_{+}^{∨}) represented by points p ∈ X sweep out an irreducible quartic hypersurface Γ ⊂ P^{5} (called “da Palatini” by Fano). Through the generic point x ∈ Γ, passes exactly one L_{p}, with p ∈ X.

On the other hand, since B is smooth, each point q ∈ B = Gr(2, 6)^{∨} ∩ P^{4} corresponds to a hyperplane H_{q} tangent to the Grassmannian in exactly one point n_{q}, called the “centre” of H_{q}. The lines N_{q} in P^{5} represented by centres n_{q} of points q ∈ B sweep out an irreducible variety Σ ⊂ P^{5}.

It is easy to see that Σ ⊂ Γ. (See for instance [Pu], pp. 83–84, where the given argument holds whenever B is a smooth 3-fold.) Now through the generic point of Σ passes exactly one line N_{q}, with q ∈ B. Otherwise, if q, q′ ∈ B and N_{q} ∩ N_{q′} ≠ ∅, then the whole pencil spanned by q and q′ lies in B, which means that we found a line in B. But this contradicts the fact that the Fano variety of the Klein cubic is 2-dimensional. Therefore Σ ⊂ P^{5} is an irreducible hypersurface, and so we must have Σ = Γ. (The quartic equation defining Γ = Σ is the unique quartic invariant for the action of SL_{2}(Z_{11}) on P^{+} via ρ_{+}; see [Ad4], Corollary 50.2 for the explicit equation.)

Choose now a generic hyperplane Π ⊂ P^{5}, and let Γ̄ := Γ ∩ Π. We may define birational maps

η : X  Γ̄,  η(p) := Π ∩ L_{p},
γ : B  Γ̄,  γ(q) := Π ∩ N_{q}.

The composition χ := γ^{−1} ∘ η defines now a birational isomorphism between X and B, as required. (See [Pu] and [Is2] for a detailed analysis of this mapping.)

Remark 2.8 1) The birational isomorphism χ provided in the proof of Theorem 2.6 and Theorem 0.1 depends on the choice of a hyperplane Π ⊂ P^{5}, and thus is not compatible with the action of PSL_{2}(Z_{11}). The indeterminacy locus of the isomorphism χ (as well as of its inverse) turns out to be the union of an elliptic quintic curve E and 25 mutually disjoint secant lines to it (which are flopped by χ). In terms of linear systems, χ is defined by |5H − 3E|, where H is the hyperplane class on X. Similarly, χ^{−1} is induced by |7H′ − 4E′|, where H′ is the hyperplane class on B and E′ is the base locus of γ.

2) Takeuki [Ta] and Tregub [Tr] have constructed a different birational isomorphism of a smooth Fano 3-fold V_{14} of genus 8, index one onto a smooth cubic hypersurface B ⊂ P^{4}, which can be briefly described as follows. Let C be a (general) rational normal curve C on B ⊂ P^{4}. There are exactly 16 chords l_{i}, 1 ≤ i ≤ 16 to C on B. Let B̃ be the blowing-up of B along C and the l_{i}’s. Then the linear system L = |8H − 5C − ∑_{i=1}^{16} 2L_{i}| provides a birational morphism from B̃ onto the intersection X of Gr(2, 6) with a codimension 5 linear subspace. Under this morphism, the unique divisor D ∈ |3H − 2C − ∑_{i=1}^{16} L_{i}| is contracted to a point p. The inverse birational morphism is then induced by the linear system |2H′ − 3p|, where H′ is the hyperplane class on X.

3) Notice also that every (abstract) smooth Fano 3-fold of genus 8, index one is isomorphic to a codimension 5 linear section of Gr(2, 6), cf. [Gu].

Question 2.9 It seems plausible that the lines on the codimension 5 linear section Y of Gr(2, 6), which is the Zariski closure of im(Θ) in Theorem 2.6, are parametrized by the modular curve X(11), and that the intermediate Jacobian of Y is isomorphic to the generalized Prym variety corresponding to the (symmetric) Hecke correspondence T_{3} on X(11). See, for instance, [Ad2] and [Ed] for a geometric description of this Hecke correspondence.

Remark 2.10 Let (A, L) be a general (1, 11)-polarized abelian surface, where L is assumed to be symmetric. One can show that the linear system |2L − 2 ∑_{i=1}^{16} e_{i}|^{+}, of even divisors of 2L having multiplicity two in the half periods, descends to a (complete) very ample linear system on the (desingularized) Kummer surface X associated to A, and embeds it as a codimension 8 linear section of the spinor variety S ⊂ P^{15}, which parametrizes isotropic P^{4}’s in an 8-dimensional smooth quadric in P^{9}. It would be interesting, in the light of [Muk], to determine exactly which codimension 8 linear sections correspond to such Kummer surfaces.

3 Moduli of (1, 9)-polarized abelian surfaces

As mentioned in the introduction, an argument similar to the one used in Section 2 allows us to prove the rationality of A_{9}^{lev}.

For the remaining of the paper, let (A, L) be a general abelian surface with a polarization of type (1, 9) and with canonical level structure. Most of the facts concerning theta groups from Section 1 can be adapted to this case, but we will make little use of them in the sequel. We will also assume that L is chosen to be a symmetric line bundle.

As seen in Section 1 |L| embeds A ⊂ P^{8} = P(H^{0}(L)^{∨}) = P(V^{∨}) invariantly under the action of the Heisenberg group H_{9} via the Schrödinger representation, and the involution ι. In particular, H^{0}(I_{A}(n)) is a representation of weight n of the Heisenberg group, whose irreducible components will have dimension 9/ gcd(9, n). Via |L|, A is embedded as a projectively normal surface of degree 18 which is contained in 9 quadrics (cf. [Laz], or [GP1], Theorem 6.5). However, in contrast with Section 2, we are in a boundary case, in that these quadrics do not generate the homogeneous ideal of A. Moreover, in general the quadrics containing the degenerations used in the proof of Theorem 2.2 and [GP1], Theorem 6.5, b), cut out only a threefold.

As in Section 2 we will investigate the locus of odd 2-torsion points, which in this simpler case turns out to be the whole of P^{−} = P(V_{−}^{∨}) ≅ P^{3}.

The space of quadrics H^{0}(O_{P^{8}}(2)) decomposes into five 9-dimensional representations of the Heisenberg group, each one isomorphic to the Schrödinger representation. As above, one such decomposition is given by the spans of the rows of the matrix defined in [GP], §6:

R_{4} = ( \begin{matrix}x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} & x_{5}^{2} & x_{6}^{2} & x_{7}^{2} & x_{8}^{2} \\ x_{1}x_{8} & x_{2}x_{0} & x_{3}x_{1} & x_{4}x_{2} & x_{5}x_{3} & x_{6}x_{4} & x_{7}x_{5} & x_{8}x_{6} & x_{0}x_{7} \\ x_{2}x_{7} & x_{3}x_{8} & x_{4}x_{0} & x_{5}x_{1} & x_{6}x_{2} & x_{7}x_{3} & x_{8}x_{4} & x_{0}x_{5} & x_{1}x_{6} \\ x_{3}x_{6} & x_{4}x_{7} & x_{5}x_{8} & x_{6}x_{0} & x_{7}x_{1} & x_{8}x_{2} & x_{0}x_{3} & x_{1}x_{4} & x_{2}x_{5} \\ x_{4}x_{5} & x_{5}x_{6} & x_{6}x_{7} & x_{7}x_{8} & x_{8}x_{0} & x_{0}x_{1} & x_{1}x_{2} & x_{2}x_{3} & x_{3}x_{4} \end{matrix} ) .

Thus every 9-dimensional H_{9}-subrepresentation of quadrics is spanned by v · R_{4} for some v ∈ C^{5} = V_{+}, and thus these representations are parametrized by P^{+} := P(V_{+}^{∨}). If we restrict R_{4} to P^{−} = P(V_{−}^{∨}), the (−1)-eigenspace of the involution ι, and consider as before the first 5 × 5 block, we obtain the matrix

S = ( \begin{matrix}0 & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} & x_{4}^{2} \\ −x_{1}^{2} & 0 & x_{3}x_{1} & x_{4}x_{2} & −x_{4}x_{3} \\ −x_{2}^{2} & −x_{3}x_{1} & 0 & −x_{4}x_{1} & −x_{3}x_{2} \\ −x_{3}^{2} & −x_{4}x_{2} & x_{4}x_{1} & 0 & −x_{2}x_{1} \\ −x_{4}^{2} & x_{3}x_{4} & x_{2}x_{3} & x_{1}x_{2} & 0 \end{matrix} ) ,

representing the intertwining operator Φ : ∧^{2}(V_{+}) → S^{2}(V_{−}).

As in Section 2, or [GP1], §6, it follows that for a point P ∈ P^{−}, v · R_{4}(P) = 0 if and only if v · S(P) = 0.

Lemma 3.1

1) rank(S(P)) ≥ 2, for all P ∈ P^{−}.

2) The locus D_{1} ⊆ P^{−} where matrix S has rank 2 is the disjoint union of a smooth curve C ⊂ P^{−} of degree 9, which is the complete intersection

{x_{1}^{2}x_{2} − x_{2}^{2}x_{4} − x_{1}x_{4}^{2} = x_{1}x_{2}^{2} − x_{3}^{3} + x_{1}^{2}x_{4} − x_{2}x_{4}^{2} = 0} ⊂ P^{−},

and the four points

P_{1} = (0 : 0 : 0 : 1 : 0 : 0 : −1 : 0 : 0), P_{2} = (0 : −1 : 1 : 0 : −1 : 1 : 0 : −1 : 1),

P_{3} = (0 : −1 : ξ^{3} : 0 : −ξ^{6} : ξ^{6} : 0 : −ξ^{3} : 1), P_{4} = (0 : −1 : ξ^{6} : 0 : −ξ^{3} : ξ^{3} : 0 : −ξ^{6} : 1),

where ξ = e^{\frac{2πi}{9}} is a primitive root of order nine of the unity. The curve C is isomorphic to the modular curve X(9).

Proof: Direct computation and arguments similar to those used in the proof of Lemma 2.1.

As in Section 2, we may interpret D_{1} as the locus of points in P^{−} ⊂ P^{8} = P(V^{∨}) which are contained in a net of H_{9}-subrepresentations of quadrics. On the other hand, S is a 5 × 5 skew-symmetric matrix, so S drops rank on all of P^{−}. Therefore we can define again a map

Θ : P^{−} \ D_{1}  P^{+}
P^{−} \ D_{1} ∋ P  P(ker(S(P))) = P({v ∈ V_{+} | v · S(P) = 0}),

which sends a point P to the unique H_{9}-subrepresentation of H^{0}(O_{P^{8}}(2)) of quadrics containing it.

By an argument similar to the one used in the proof of Lemma 2.5, the morphism Θ is easily seen to map a point P ∈ P^{−} to the point of P^{+} whose coordinates are given by the 4×4-Pfaffians of S(P), taken with suitable signs. In coordinates, if P = (x_{1} : …: x_{4}) ∈ P^{−} this yields Θ(x_{1} : …: x_{4}) = (v_{0} : …: v_{4}), where

\begin{aligned}v_{0} & = −x_{1}^{2}x_{2}x_{3} + x_{2}^{2}x_{3}x_{4} + x_{1}x_{3}x_{4}^{2} \\ v_{1} & = x_{1}x_{2}^{3} − x_{2}x_{3}^{3} + x_{1}x_{4}^{3} \\ v_{2} & = −x_{1}^{3}x_{2} + x_{3}^{3}x_{4} + x_{2}x_{4}^{3} \\ v_{3} & = x_{1}^{2}x_{2}x_{3} − x_{2}^{2}x_{3}x_{4} − x_{1}x_{3}x_{4}^{2} \\ v_{4} & = x_{1}x_{3}^{3} − x_{1}^{3}x_{4} − x_{2}^{3}x_{4} \end{aligned} .

Since v_{0} = −v_{3}, we deduce that the image of Θ is contained in the linear subspace Π of P^{+} defined by v_{0} = −v_{3}. As in Section 2, by [GP1], Lemma 6.4, Θ induces a rational map

Θ_{9} : A_{9}^{lev} Π,

which essentially is defined by taking an abelian surface A ⊆ P^{8} to Θ((A ∩ P^{−}) \ D_{1}), that is to the point corresponding to the unique H_{9}-subrepresentation of H^{0}(O_{P^{8}}(2)) of quadrics containing the abelian surface.

Remark 3.2 It is easy to see that a (1, 9)-polarized abelian surface A ⊆ P^{8} is not cut out by quadrics. Indeed, if v = (v_{0} : …: v_{4}) = Θ_{9}(A) ∈ im(Θ), then v_{0} = −v_{3} and each quadric entry of v · R_{4} vanishes at the (fixed) point

P = (1 : 0 : 0 : 1 : 0 : 0 : 1 : 0 : 0) ∈ P^{8}.

However, since σ^{3}(P) = P and τ^{3}(P) = P, where σ and τ acting by translation by 9torsion points on A are the usual generators of H_{9} in the Schrödinger representation, P cannot be contained in A. Thus A is not cut out by quadrics since the only quadrics containing A are linear combinations of the entries of v · R_{4}. In fact one may show that for the general abelian surface A, the quadrics defined by v · R_{4} cut out the union of A and the set of nine points which form the H_{9} orbit of P. A degeneration argument, in the spirit of [GP1], §6, shows that the homogeneous ideal of A is in fact generated by the 9 quadrics and 6 extra cubics (use Lemma 3.4 below).

We can now prove the rationality of A_{9}^{lev}:

Theorem 3.3 Θ_{9} : A_{9}^{lev} Π ≅ P^{3} is a birational map.

Proof: We follow much the same strategy as the proof of [GP1], Theorem 6.5, however since, by Remark 3.2, quadrics do not cut out an abelian surface, we will need to involve cubic equations, and the process is a bit more difficult computationally.

We will also make use of the ubiquitous (Moore) 9 × 9-matrices

M′_{4}(x, y) = (x_{5(i+j)}y_{5(i−j)})_{i,j∈Z_{9}},

where we think of x = (x_{i})_{i∈Z_{9}} as a point in the ambient P^{8} and y = (y_{i})_{i∈Z_{9}} as a parameter point. We refer the reader to [GP1], §2 and §6 for a detailed discussion of their properties. Note also that the matrix R_{4} above, up to transpose and permutations of rows and columns, is a submatrix of M′_{4}(x, x).

Let Z := Θ^{−1}(im(Θ_{9})) ⊆ P^{−} \ D_{1}, and let Z̄ denote the closure of Z in P^{−} \ D_{1}. Let A ⊆ P8 ; Z̄ be the family defined by the condition that the ideal of a fibre A_{z}, z ∈ Z̄, is generated by the 9 quadrics, which are entries of Θ(z) · R_{4} (i.e., the H_{9}-subrepresentation of H^{0}(O_{P^{8}}(2)) vanishing at z), along with the cubics which are the 6 × 6-Pfaffians of the skew-symmetric 9 × 9-matrix M′_{4}(x, z). By [GP1], Corollary 2.8 and Lemma 6.4, A_{z} contains all abelian surfaces whose odd 2-torsion points map to z.

We need to show that there exists an open set U ⊂ Z̄, such that the restricted family A_{U} U is flat, and every smooth fiber is an H_{9}-invariant (and thus (1, 9)-polarized) abelian surface.

The degeneration argument in [GP1], Theorem 3.1 and Theorem 6.2 shows that if E ⊂ P^{8} is a Heisenberg invariant elliptic normal curve of degree 9, then Sec(E)∩(P^{−}\D_{1}) ⊆ Z̄. The same is also true if we take E to be the “standard 9-gon” X(Γ_{9}), and Sec(E) to be its “secant variety”, that is, with notation as in [GP1], §4:

X(Γ_{9}) = ∪_{i∈Z_{9}}l_{i,i+1} ⊆ P^{8},

where l_{i,i+1} = ⟨e_{i}, e_{i+1}⟩ is the line joining the vertices e_{i} and e_{i+1} of the standard simplex in P^{8}. In particular, for E = X(Γ_{9}), the set Sec(E) ∩ (P^{−} \ D_{1}) and thus also Z̄ contain the point

z_{0} = (0 : 0 : −1 : −1 : 0 : 0 : 1 : 1 : 0).

Let I_{0} be the homogeneous ideal of the fibre A_{z_{0}}. To conclude the result, it will be enough to show that A_{z_{0}} is contained in a surface of degree 18.

Now Θ(z_{0}) = (0 : 1 : 0 : 0 : 0) ∈ Π ⊂ P^{+}, so I_{0} contains the quadrics Θ(z) · R_{4}, namely:

{x_{i}x_{i+2}, i ∈ Z_{9}}.

On the other hand, the matrix M′_{9}(x, z_{0}) is

( \begin{matrix}0 & 0 & 0 & −x_{6} & x_{2} & −x_{7} & x_{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & −x_{7} & x_{3} & −x_{8} & x_{4} & 0 \\ 0 & 0 & 0 & 0 & 0 & −x_{8} & x_{4} & −x_{0} & x_{5} \\ x_{6} & 0 & 0 & 0 & 0 & 0 & −x_{0} & x_{5} & −x_{1} \\ −x_{2} & x_{7} & 0 & 0 & 0 & 0 & 0 & −x_{1} & x_{6} \\ x_{7} & −x_{3} & x_{8} & 0 & 0 & 0 & 0 & 0 & −x_{2} \\ −x_{3} & x_{8} & −x_{4} & x_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & −x_{4} & x_{0} & −x_{5} & x_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & −x_{5} & x_{1} & −x_{6} & x_{2} & 0 & 0 & 0 \end{matrix} )

We consider first two of its 6 × 6-Pfaffians: The skew-symmetric 6 × 6-minor coming from taking rows (and columns) 1,2,3,5,6 and 7 has Pfaffian

−x_{2}x_{3}x_{4} + x_{4}x_{7}^{2} − x_{3}x_{7}x_{8} + x_{2}x_{8}^{2} ∈ I_{0},

and similarly taking rows (and columns) 1,2,3,4,6, and 8, we get another cubic Pfaffian

−x_{0}x_{3}x_{6} + x_{4}x_{6}x_{8} ∈ I_{0}.

Taking into account the quadrics in I_{0}, we observe that I_{0} also contains the polynomials x_{4}x_{7}^{2} − x_{3}x_{7}x_{8} + x_{2}x_{8}^{2} and x_{0}x_{3}x_{6}. Since the matrix M′_{9} is Heisenberg invariant (in the x-coordinate) up to permutations of rows and columns, it follows that I_{0} is H_{9}-invariant, and hence contains

x_{i}x_{i+2}, i ∈ Z_{9}
x_{0}x_{3}x_{6}, x_{1}x_{4}x_{7}, x_{2}x_{5}x_{8},
x_{i+4}x_{i+7}^{2} − x_{i+3}x_{i+7}x_{i+8} + x_{i+2}x_{i+8}^{2}, i ∈ Z_{9}.

The claim of Theorem 3.3 follows now from the following combinatorial lemma, which determines the Hilbert polynomial of I_{0}:

Lemma 3.4

1) The ideal J_{1} generated by the quadric and cubic monomials

{x_{i}x_{i+2}, x_{i}x_{i+3}x_{i+6}, x_{i+3}x_{i+7}x_{i+8} | i ∈ Z_{9}}

is the Stanley-Reisner face ideal I_{X(Δ_{9})} corresponding to the triangulation Δ_{9} of the torus T_{1} in [GP1], Proposition 4.4. In particular, J_{1} has the same Hilbert polynomial as a (1, 9)-polarized abelian surface.

2) The ideal J_{2} generated by the 12 quadric and cubic monomials

{x_{i}x_{i+2}, x_{i}x_{i+3}x_{i+6} | i ∈ Z_{9}},

cuts out the threefold

Σ = ⋃_{i=0}^{8} L_{i}, L_{i} = σ^{i}(L_{0}),

where L_{0} is the P^{3} determined by {x_{0} = x_{1} = x_{4} = x_{5} = x_{8} = 0}. J_{2} is the face ideal of the “solid” torus whose triangulation Δ_{9} is described in 1). This is the complex whose two-simplices are those of Δ_{9} but which has in addition three-simplices with vertices (x_{i}, x_{i+1}, x_{i+3}, x_{i+4}).

3) The ideals

J_{(λ:μ)} = J_{2} + ⟨λx_{i+4}x_{i+7}^{2} − μx_{i+3}x_{i+7}x_{i+8} + λx_{i+2}x_{i+8}^{2}, i ∈ Z_{9}⟩,

for (λ : μ) ∈ P^{1}, define a flat family of surfaces X_{(λ:μ)} ⊂ P^{8} with the same Hilbert polynomial as a (1, 9)-polarized abelian surface. In particular, I_{0} = J_{(1:1)} defines a surface of degree 18 as desired.

Proof: The proof is easy and left to the reader. Observe that X_{(λ:μ)} is defined by J_{2} and 9 trinomials, from which it can be shown that set theoretically X_{(λ:μ)} is the union of 9 distinct (smooth) quadric surfaces

X_{(λ:μ)} = ⋃_{i=0}^{8} Q_{i}, Q_{i} = σ^{i}(Q_{0}),

where Q_{0} is defined by

Q_{0} = L_{0} ∩ {λx_{3}x_{6} − μx_{2}x_{7} = 0}.

On the other hand, J_{(0:1)} = J_{1} is the Stanley-Reisner ideal of the triangulation Δ_{9} of the torus, and thus has the required Hilbert function. (See also [GP1], Proposition 4.4, for details.)

Remark 3.5 The linear projection π_{−} : P(V^{∨}) = P^{8} P^{−} commutes with the involution ι and thus maps a general H_{9} ⋊⟨ι⟩ abelian surface A ⊂ P^{8} to a 6-nodal Kummer quartic surface K ⊂ P^{3} = P^{−} (whose nodes are the odd 2-torsion points of A). The linear system of quadrics through the set S of nodes of K maps the Kummer surface to a smooth quartic K′ ⊂ P^{3}. Such a smooth quartic surface has 16 skew conics, and in fact any smooth quartic surface in P^{3} containing 16 skew conics is a Kummer surface of an abelian surface (A, L) with a polarization of type (1, 9), via the linear system |2L − 2 ∑_{i=1}^{16} e_{i}|^{+} of even divisors of the totally symmetric line bundle 2L, having multiplicity two in the half periods (see [BB], Claims 2–4, and [Bau], Theorem 2.1 for a detailed discussion).

Remark 3.6 There is a second family of (minimal) abelian surfaces of degree 18, and sectional genus 10 embedded in P^{8}, namely those embedded via a polarization of type (3, 3). These are also contained in 9 independent quadrics, that, in contrast with the (1, 9) case, cut out scheme theoretically the abelian surface. The homogeneous ideal of a (3, 3) polarized abelian surface is generated by (quadrics and three independent) cubics (cf. [Se]). See [Co], [Gra], [vdG], and [Ba] for explicit equations and their relation to the Burchardt quartic.

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