The Moduli Space of Bilevel-6 Abelian Surfaces

X iv :m at h/ 00 12 20 2v 1 [ m at h. A G ] 2 0 D ec 2 00 0 The Moduli Space of Bilevel-6 Abelian Surfaces G.K. Sankaran & J. Spandaw The moduli space Abil t of (1, t)-polarised abelian surfaces with a weak bilevel structure was introduced by S. Mukai in [11]. Mukai showed that Abil t is rational for t = 2, 3, 4, 5. More generally, we may ask for birational invariants, such as Kodaira dimension, of a smooth model of a compactification of Abil t : since the choice of model does not affect birational invariants, we refer to the Kodaira dimension, plurigenera, etc., of Abil t . From the description of Abil t as a Siegel modular 3-fold Γ t \H2 and the fact that Γ t ⊂ Sp(4,Z) it follows, by a result of L. Borisov [3], that κ(Abil t ) = 3 for all sufficiently large t. For an effective result in this direction see [10]. In this note we shall prove an intermediate result for the case t = 6. Theorem. The moduli space Abil 6 has geometric genus pg(A 6 ) ≥ 2 and Kodaira dimension κ(Abil 6 ) ≥ 1. The case t = 6 attracts attention for two reasons: it is the first case not covered by the results of [11]; and the Humbert surface H1(1) ⊂ Abil t , which in the cases 2 ≤ t ≤ 5 is a quadric and plays an important role both in [11] and below, becomes an abelian surface (at least birationally) because the modular curve X(6) has genus 1. The method we use is that of Gritsenko, who proved a similar result for the moduli spaces of (1, t)polarised abelian surfaces with canonical level structure for certain values of t: see [6], especially Corollary 2. We use some of the weight 3 modular forms constructed by Gritsenko and Nikulin as lifts of Jacobi forms in [9] to produce canonical forms having effective, nonzero, divisors on a suitable projective model X6 of Abil 6 . A similar method was applied to the Barth–Nieto threefold by Gritsenko and Hulek in [8]. Acknowledgements: We are grateful to the DAAD and the British Council for financial assistance under ARC Project 313ARC-XIII-99/45.