Formal Brauer groups and the moduli of abelian surfaces

Let X be an algebraic surface over an algebraically closed field k of characteristic p > 0. We denote by ΦX the formal Brauer group of X and by h = h(ΦX) the height of ΦX . In a previous paper, [6], we examined the structure of the stratification given by the height h in the moduli space of K3 surfaces, and we determined the cohomology class of each stratum. In this paper, we apply the methods of [6] to treat the case of abelian surfaces. In this case, the situation is more concrete, and so we can more easily determine the structure of the stratification given by the height h(ΦA) in the moduli of abelian surfaces. For the local structure we refer to [19]. On the moduli of principally polarized abelian varieties in positive characteristic there is another natural stratification, called the Ekedahl-Oort stratification, cf. [13], and one can calculate the corresponding cycle classes [5]. Although our three strata coincide set-theoretically with strata of the Ekedahl-Oort stratification, there is a subtle difference: one of the strata comes with multiplicity 2. We will here summarize our results. We consider the moduli stack M = A2 of principally polarized abelian surfaces over k; alternatively, we can consider the moduli spaces M = A2,n (n ≥ 3, p 6 |n) of principally polarized abelian surfaces with level n-structure. We know that M is a 3-dimensional algebraic stack (variety). We let π : X → M be the universal family over M . We set M (h) := {s ∈ M : h(ΦXs) ≥ h}. Note that M (3) = M (∞). The moduli stack M possesses a natural compactification M̃ which is an example of a smooth toroidal compactification, cf.