Classification of Horikawa surfaces with T-singularities

TL;DR

This paper classifies all projective surfaces with T-singularities and specific invariants, revealing their smoothability properties and providing new insights into the structure of Horikawa surfaces within the KSBA moduli space.

Contribution

It provides a comprehensive classification of T-singularities on Horikawa surfaces and analyzes their smoothability, introducing new techniques for classifying KSBA surfaces with T-singularities.

Findings

Surfaces with $p_g \, \geq \, 10$ are generally not smoothable, except for specific Lee-Park examples.

Most Lee-Park surfaces have a unique deformation type unless $p_g=6$, where two types exist.

The classification extends to surfaces with $K^2 \leq 2p_g - 3$, including quintic and I-surfaces.

Abstract

We classify all projective surfaces with only T-singularities, ample canonical class, and $K^2=2p_g-4$. In this way, we identify all surfaces, smoothable or not, with only T-singularities in the Koll\'ar--Shepherd-Barron--Alexeev (KSBA) moduli space of Horikawa surfaces. We also prove that they are not smoothable when $p_g \geq 10$, except for the Lee-Park (Fintushel-Stern) examples, which we show to have only one deformation type unless $p_g=6$ (in which case they have two). This demonstrates that the challenging Horikawa problem cannot be addressed through complex T-degenerations. We propose new questions regarding diffeomorphism types based on our classification. Furthermore, the techniques developed in this paper enable us to classify all KSBA surfaces with only T-singularities and $K^2\leq 2p_g-3$, for example, quintic surfaces and I-surfaces.