Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

TL;DR

This paper constructs and studies new families of simply-connected surfaces with ample canonical class and geometric genus one, relating them to rational curve configurations in K3 surfaces through $ ext{Q}$-Gorenstein smoothings, including the first known examples for certain invariants.

Contribution

It establishes the existence of new families of simply-connected surfaces with specific invariants and links them to rational curve configurations in K3 surfaces, including the first examples for $K^2=7$ and $K^2=9$.

Findings

Existence of $(20-2K^2)$-dimensional families of surfaces for $1 \, ext{to}\, 9$.

First known surfaces with $K^2=7$ and $K^2=9$.

A 4-dimensional family of surfaces for $K^2=8$.

Abstract

We prove the existence of $(20-2K^2)$-dimensional families of simply-connected surfaces with ample canonical class, $p_g=1$, and $1 \leq K^2 \leq 9$, and we study the relation with configurations of rational curves in K3 surfaces via $\mathbb Q$-Gorenstein smoothings. Our surfaces with $K^2=7$ and $K^2=9$ are the first surfaces known in the literature, together with the existence of a $4$-dimensional family for $K^2=8$.