Counting Lines with Vinberg's algorithm

TL;DR

The paper develops a two-pronged framework for counting and classifying lines on complex K3 surfaces, extending methods from smooth cases to singular models with Du Val singularities. By combining Vinberg's algorithm with lattice-theoretic/arithmetic techniques, it analyzes large line configurations via extended Fano graphs and NS-lattice considerations, and applies the method to degree‑8 K3 octics to derive sharp bounds and enumerate extremal families. A key finding is that a larger NS-lattice can yield fewer lines, contrasting with the smooth case and informing the use of partitions into lines and exceptional divisors. The results include a precise bound of $36$ lines (or $32$ in the singular case) and explicit classifications of Kummer octics, special octics, and related triquadrics, with computational components underpinning the classification.

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.

Theorems & Definitions (2)

• Theorem 1.1: see \ref{['S.Proof']}
• Theorem 1.2: see \ref{['proof.special']}