Invariants of compactified moduli spaces of polarized abelian surfaces (Q2785961)

The moduli space \({\mathcal A}_{1,p}\) of polarized abelian surfaces of type \((1,p)\) with canonical level structure is a quotient \(H^2/ \Gamma\) of the Siegel upper half space \(H^2\) by an arithmetic subgroup \(\Gamma\) of \(\text{Sp}_4 (\mathbb{Q})\). It is a 3-dimensional quasi-projective variety. Its toroidal compactification \({\mathcal A}^*_{1,p}\) admits a natural desingularization \(\widetilde {\mathcal A}_{1,p}\). In the paper under review invariants of \(\widetilde {\mathcal A}_{1,p}\) such as the canonical divisor and its self-intersection are computed. The study of divisors and intersection numbers allows to give a lower bound of the Picard number of \(\widetilde {\mathcal A}_{1,p}\). The results provide for instance a first step in the construction of a minimal model in the sense of Mori theory.