Standard stable Horikawa surfaces

TL;DR

This paper studies the compactification of the moduli space of Horikawa surfaces, revealing intersections of components and the emergence of non-reduced components as the canonical volume increases.

Contribution

It provides an explicit description of the boundary of the moduli space and analyzes the structure of its components for surfaces with specific invariants.

Findings

Intersections of moduli space components for certain invariants.

Explicit description of semi-smooth surfaces in the boundary.

Increase in non-reduced components with larger invariants.

Abstract

We consider the stable compactification $\bar {\mathfrak H}$ of the moduli space of Horikawa surfaces with $K_X^2 = 2p_g(X) -4$. When $K_X^2 =8\ell$ we show that the closures of the two components $\mathfrak H^{\mathrm I}$ and $\mathfrak H^{\mathrm {II}}$ of the Gieseker moduli space intersect, for $\ell>2$ in a divisor parametrising explicitly described semi-smooth surfaces. With growing $K_X^2$ we find an increasing number of generically non-reduced irreducible components in the same connected component of the moduli space of stable surfaces.