Lines on K3-sextics with simple singularities

TL;DR

This paper classifies line configurations on K3-sextic surfaces with simple singularities, revealing new automorphism group structures and excluding certain line arrangements, thereby advancing understanding of K3 surface geometry.

Contribution

It provides a comprehensive classification of lines on K3-sextics with A-D-E singularities and characterizes infinite dihedral automorphism groups, a novel insight in K3 surface theory.

Findings

Classified configurations of at least 36 lines on K3-sextics

Characterized infinite dihedral groups of automorphisms

Proved no K3-sextic contains a Kummer configuration of lines

Abstract

We advance our understanding of the configurations of low degree smooth rational curves on (quasi-)polarized complex K3-surfaces. We apply our efficient approach to classify the configurations of at least 36 lines on K3-sextics with at worst A-D-E singularities. As an unexpected outcome of the further analysis of configurations of lines, we characterize a certain class of infinite dihedral groups of birational automorphisms of K3-sextics. Besides, we show that no K3-sextic can contain a Kummer configuration of lines, and we give a complete account of the line configurations on closest analogue of Kummer K3-octics or quartics, viz. the so-called Humbert K3-sextics.