Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

TL;DR

This paper explores the boundary structure of the moduli space of certain algebraic surfaces, identifying new boundary divisors and analyzing their relation to GIT compactification and Hodge theory.

Contribution

It introduces eight new irreducible boundary divisors in the KSBA compactification of the moduli space of specific Horikawa surfaces, linking boundary geometry with GIT and Hodge theory.

Findings

Identification of eight new boundary divisors in the moduli space.

Analysis of the relation between boundary divisors and GIT compactification.

Insights into the Hodge theory of degenerate surfaces in the boundary.

Abstract

Smooth minimal surfaces of general type with $K^2=1$, $p_g=2$, and $q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space $\mathbf{M}$ of their canonical models admits a modular compactification $\overline{\mathbf{M}}$ via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of $\mathbf{M}$ and the Hodge theory of the degenerate surfaces that the eight divisors parametrize.