Deformations of I-surfaces with elliptic singularities

TL;DR

This paper studies I-surfaces of general type with specific singularities, providing a detailed description of their deformation theory and clarifying how their moduli space can be explicitly characterized.

Contribution

It offers a precise analysis of I-surfaces with elliptic singularities, including their deformation behavior and exceptions, advancing understanding of their moduli space.

Findings

Deformations are versal for simple elliptic and cusp singularities under mild conditions.

Two specific exceptions where deformation behavior differs are identified and analyzed.

Provides a detailed description of the deformation theory for these I-surfaces.

Abstract

An I-surface $S$ is an algebraic surface of general type with $K_S^2 = 1$ and $p_g(S) = 2$. Recent research has centered on trying to give an explicit description of the KSBA compactification of the moduli space of these surfaces. The possible normal Gorenstein examples have been enumerated by work of Franciosi-Pardini-Rollenske. The goal of this paper is to give a more precise description of such surfaces in case their singularities are simple elliptic and/or cusp singularities, and to work out their deformation theory. In particular, under some mild general position assumptions, we show that deformations of the surfaces in question are versal for deformations of the singular points, with two exceptions where the discrepancy is analyzed in detail.