Rational homology disk degenerations of elliptic surfaces

TL;DR

This paper classifies and constructs rational homology disk degenerations of elliptic surfaces, extending previous classifications and demonstrating unobstructed deformations to recent Dolgachev surface degenerations.

Contribution

It extends Kawamata's classification to all $ ext{Q}$HD degenerations of elliptic surfaces and constructs explicit minimal slc models for these degenerations.

Findings

Classified all $ ext{Q}$HD$ degenerations of elliptic surfaces.

Constructed minimal slc models via Seifert resolutions and flips.

Proved unobstructed deformations to recent Dolgachev surface degenerations.

Abstract

In this paper, a $\mathbb{Q}$HD singularity is a weighted homogeneous normal surface singularity admitting a rational homology disk ($\mathbb{Q}$HD) smoothing. These singularities are rational but often not log canonical. We classify all $\mathbb{Q}$HD degenerations of nonsingular projective elliptic surfaces, extending Kawamata's classification of the case with only Wahl singularities (i.e., log terminal $\mathbb{Q}$HD singularities). We also realize all $\mathbb{Q}$HD degenerations of Dolgachev surfaces $D_{a,b}$ with one $\mathbb{Q}$HD singularity, for every pair of integers $a,b$. For each such degeneration, we construct a minimal semi log canonical (slc) birational model via a Seifert partial resolution in the sense of Wahl followed by semistable flips. Finally, we prove that these minimal slc models are unobstructed and deform to the recent degenerations of Dolgachev surfaces constructed by D. Lee and Y. Lee.