Involutions, Humbert Surfaces, and Divisors on a Moduli Space

Let M2 be the Igusa compactification of the Siegel modular variety of degree 2 and level 2. In earlier work with R. Lee, we carefully investigated this variety. Subvarieties Dl (compactification divisors) and H∆ (Humbert surface of discriminant 1) play a prominent role in its structure; in particular their fundamental classes span H4(M2;Z). We return to this variety and consider another class of subvarieties Kh (Humbert surfaces of degree 4), which we investigate with the help of involutions on M2. We carefully describe these subvarieties and consider the representations of their fundamental classes in terms of the fundamental classes of the subvarieties Dl and H∆. The space M2 is also known in a different context. It can also be described as the space M0,6 of stable curves of genus 2 with ordered Weierstrass points. In this context the divisors Kh are what have come to be known as Keel-Vermeire divisors. Let S2 denote Siegel space of degree two, i.e. the space of symmetric 2-by-2 complex matrices with positive definite imaginary part. Let Sp(4,Z) be the group of 4-by-4 matrices that preserves the usual symplectic form on Z4. Then Sp(4,Z) acts on S2 on the left by