On some boundary divisors in the moduli spaces of stable Horikawa surfaces with $K^2=2p_g-3$

TL;DR

This paper characterizes certain boundary divisors in the moduli space of stable Horikawa surfaces with specific invariants, revealing their geometric properties and singularities, and establishing their relation to the Gieseker moduli space.

Contribution

It provides a detailed description of boundary divisors in the moduli space of stable Horikawa surfaces with K^2=2p_g-3, including their singularities and intersection properties.

Findings

General points of the divisor D correspond to surfaces with a specific cyclic quotient singularity.

The divisor D is contained in the closure of the Gieseker moduli space and intersects all its components.

The moduli space is smooth at a general point of D.

Abstract

We describe the normal stable surfaces with K^2=2p_g-3 and p_g>14 whose only non canonical singularity is a cyclic quotient singularity of type 1/4k(1,2k-1) and the corresponding locus D inside the KSBA moduli space of stable surfaces. More precisely, we show that: (1) a general point of any irreducible component of D corresponds to a surface with a singularity of type 1/4(1,1), (2) the closure of D is a divisor contained in the closure of the Gieseker moduli space of canonical models of surfaces with K^2=2p_g-3 and intersects all the components of such closure, and (3) the KSBA moduli space is smooth at a general point of D. In addition, we show that D has 1 or 2 irreducible components, depending on the residue class of p_g modulo 4.