Moduli of Polarised Abelian Surfaces

Abelian surfaces over C with a polarisation of type (1, t), t a positive integer, are parametrised by a coarse moduli space At which is a quasiprojective variety. In this paper we shall concentrate on the case where t is a prime p ≥ 5, and show that for p sufficiently large (in fact p ≥ 173) any algebraic compactification of Ap is of general type. A few results similar to this are already known. O’Grady, in [O’G], considers the case t = p and shows that a compactification of Ap2 is of general type for p ≥ 17 (improved to p ≥ 11 in [GS]). The special feature here is the existence of a finite morphism from Ap2 to the moduli space of principally polarised abelian surfaces (the case t = 1). This is also the case in [Bor], where it is shown that a compactification of a Siegel modular threefold coming from a subgroup Γ < Sp(4,Z) of finite index is of general type except for finitely many Γ. Another moduli space, referred to in this paper as A p and parametrising abelian surfaces with a polarisation of type (1, p) and a level structure, is studied in depth by Hulek, Kahn and Weintraub in the book [HKW2]. Its singularities are described in [HKW1] and it has been shown, by Hulek, Gritsenko and me, that it is of general type if the prime p is at least 37: see [HS] and [GH]. There is a finite morphism A p → Ap. This morphism, the singularities of Ap and its toroidal compactifications have been studied in [Br] by Brasch, who gives an analysis in the spirit of [HKW2]. Our main tools are Brasch’s results, the calculations relating to A p found in [HKW1], [HKW2] and [HS], and some special cusp forms constructed by Gritsenko (see [G]). In principle we do not need to know about A p but, like Brasch, we