Counting lines with Vinberg’s algorithm

Abstract

We combine classical Vinberg’s algorithms with the lattice-theoretic/arithmetic approach from Degtyarev (2019) 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.