Involutions, Humbert surfaces, and divisors on a moduli space

Abstract

Let M
*2 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_Δ (Humbert surface of discriminant 1) play a prominent role in its structure; in particular their fundamental classes span _H_4(M
*2; 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 M
*2. 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_Δ. The space M
*2 is also known in a different context. It can also be described as the space _M_0; 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.