Counting Lines with Vinberg's algorithm

TL;DR

This paper introduces a novel method combining Vinberg's algorithms with lattice theory to classify large line configurations on complex K3-surfaces, achieving bounds on the number of lines.

Contribution

It presents a new classification technique for line configurations on K3-surfaces using an integrated algorithmic and lattice-theoretic approach.

Findings

Classified all complex K3-octic surfaces with certain singularities and at least 32 lines.

Established an upper bound of 36 lines on these surfaces.

Identified a maximum of 32 lines when singularities are present.

Abstract

We combine classical Vinberg's algorithms with the lattice-theoretic/arithmetic approach from arXiv:1706.05734 [math.AG] to give a method of classifying large line configurations on complex quasi-polarized K3-surfaces. We apply our method to classify all complex K3-octic surfaces with at worst Du Val singularities and at least 32 lines. The upper bound on the number of lines is 36, as in the smooth case, with at most 32 lines if the singular locus is non-empty.