KSBA surfaces with elliptic quotient singularities, π1 = 1, p g = 0, and K 2 = 1, 2

Abstract

Among log canonical surface singularities, those which have a rational homology disk smoothing are the cyclic quotient singularities \(\frac{1}{{{n^2}}}\left( {1,na - 1} \right)\) with gcd(a, n) = 1, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with π1 = 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 π1 = 1, p g = 0, and K 2 = 1, 2, 3, 4.