Rational configurations in K3 surfaces and simply-connected <i>p<sub>g</sub></i>=1 surfaces with <i>K</i><SUP>2</SUP>=1, 2, 3, 4, 5, 6, 7, 8, 9
No Thumbnail Available
Date
2022
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
The aim of this paper is to directly relate the problem of constructing simply-connected p(g) = 1 surfaces of general type via Q-Gorenstein smoothings with geographical problems about arrangements of rational curves in K3 surfaces. We focus on simple normal crossing arrangements, showing numerical constraints based on their log Chern numbers and their rank in the Neron-Severi group. We also show what to expect as Wahl singularities, classifying the case when K-2 = 1. We use these arrangements to prove the existence of many (20 - 2K(2))-dimensional families of simply-connected surfaces with ample canonical class, p(g) = 1, and 1 <= K-2 <= 9, through several constructions with various special properties. We work out all our surfaces from one particular arrangement in a specific K3 surface with maximal Picard number. 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. With this, unobstructed simply-connected p(g) = 1 surfaces have been found for every possible K-2.