Involutions, Humbert surfaces, and divisors on a moduli space

Abstract

Let ${\mathfrak{M}}_2^{\ast }$ 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 $D_{\ell }$ (compactification divisors) and $H_{\Delta }$ (Humbert surface of discriminant 1) play a prominent role in its structure; in particular their fundamental classes span $H_4({\mathfrak{M}}_2^{\ast };\mathbf{Z})$. We return to this variety and consider another class of subvarieties $K_h$ (Humbert surfaces of degree 4), which we investigate with the help of involutions on ${\mathfrak{M}}_2^{\ast }$. We carefully describe these subvarieties and consider the representations of their fundamental classes in terms of the fundamental classes of the subvarieties $D_{\ell }$ and $H_{\Delta }$. The space ${\mathfrak{M}}_2^{\ast }$ is also known in a different context. It can also be described as the space ${\overline{M}}_{0,6}$ of stable curves of genus 2 with ordered Weierstrass points. In this context the divisors $K_h$ are what have come to be known as Keel–Vermeire divisors.