KSBA surfaces with elliptic quotient singularities, ${\pi }_1=1$, $p_g=0$, and $K^2=1,2$

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 $K^2=1,2,3,4$.