Normal stable degenerations of Noether-Horikawa surfaces

TL;DR

This paper classifies all normal stable Horikawa surfaces with specific singularities, provides criteria for their smoothability, and describes their moduli space boundary, advancing understanding of their geometric structure.

Contribution

It offers a complete classification of certain stable Horikawa surfaces, including smoothability criteria and boundary descriptions of their moduli space.

Findings

Classification of all such surfaces with specific singularities

Criteria for global $ ext{Q}$-Gorenstein smoothability

Description of boundary strata in the moduli space

Abstract

We classify all normal stable Horikawa surfaces with only $\mathbb{Q}$-Gorenstein smoothable log canonical singularities. Furthermore, we provide a criterion for their global $\mathbb{Q}$-Gorenstein smoothability and describe the boundary strata of the moduli space of $\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surfaces.