F eb 1 99 9 The moduli space of ( 1 , 11 ) - polarized abelian surfaces is unirational

We prove that the moduli space Alev 11 of (1, 11)-polarized abelian surfaces with level structure of canonical type is birational to Klein’s cubic hypersurface in P. Therefore, Alev 11 is unirational but not rational, and there are no Γ11-cusp forms of weight 3. The same methods also provide an easy proof of the rationality of Alev 9 . 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 Γ11-cusp forms of weight 3. Let Ad denote the moduli space of polarized abelian surfaces of type (1, d), and letAlev d be the moduli space of (1, d)-polarized abelian surfaces with canonical level structure. The map which forgets the level structure representsAlev d as a finite cover ofAd. Its general fiber is Γd/Γ lev d ∼= SL2(Zd), where Γd and Γ d 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(d2− 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. 2 M. Gross and S. Popescu Our main result is the following Theorem 0.1 The moduli space Alev 11 is birational to Klein’s cubic hypersurface K = V ( ∑ i∈Z5 xixi+1 = 0) ⊂ P. In particular, Alev 11 is unirational but not rational. The cubic hypersurface K ⊂ P 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 PSL2(Z11)-invariant of degree three in P, and that furthermore PSL2(Z11) 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: At is not unirational (and in fact pg(Ãt) ≥ 1) if t ≥ 13 and t 6= 14, 15, 16, 18, 20, 24, 30, 36 (Gritsenko [Gri1], [Gri2]), while Ãlev p is a 3-fold of general type for all primes numbers p ≥ 37 (HulekSankaran [HS1], Gritsenko-Hulek, appendix to [Gri1]), where Ãlev p is a smooth projective model of a compactification of Alev p . 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, Alev 5 ∼= P(H(FHM (3))), where bar stands for the Igusa (=Voronoi) toroidal compactification and FHM is the Horrocks-Mumford bundle on P 4 ([HM], [HKW]), while Alev 7 is rational having as birational model a smooth V22, 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 Alev 11 . In a series of forthcoming papers [GP2], [GP3], we will give details as to the structure ofAlev d , 6 ≤ d ≤ 12, (excluding d = 9 and 11, which are covered here) and Ad, d = 14, 16, 18 and 20. In particular, we will prove their rationality or unirationality. Finally, the methods used in this paper also provide an easy proof of the rationality (over Q(ξ) for ξ a primitive 9 root of unity) of Alev 9 . The unirationality of this space also follows implicitly from O’Grady’s work [O’G]. He identifies Alev p , for p prime, with the moduli space A1(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 A1(p) to the moduli space A1 of principally polarized abelian surfaces. Not all singularities of the toroidal compactification of A1(p) are canonical, so O’Grady needs to The moduli space of (1, 11)-polarized abelian surfaces is unirational 3 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.