KSBA surfaces with elliptic quotient singularities, $\pi_1=1$, $p_g=0$, and $K^2=1,2$

TL;DR

This paper constructs and classifies KSBA surfaces with specific elliptic quotient singularities, focusing on those with trivial fundamental group, geometric genus zero, and small self-intersection numbers, expanding the known catalog of such surfaces.

Contribution

It demonstrates the existence of smoothable KSBA surfaces with particular elliptic quotient singularities and provides a comprehensive list of related surface singularities with specified invariants.

Findings

Existence of smoothable KSBA surfaces with $ ext{π}_1=1$, $p_g=0$, $K^2=1,2$

Classification of surface singularities with these invariants

New examples of normal surface singularities in KSBA surfaces

Abstract

Among log canonical surface singularities, the ones which have a rational homology disk smoothing are the cyclic quotient singularities $\frac{1}{n^2}(1,na-1)$ with gcd$(a,n)=1$, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with $\pi_1=1$, $p_g=0$, and $K^2=1,2$ for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants $\pi_1=1$, $p_g=0$, and $K^2=1,2,3,4$.