On some lattice computations related to moduli problems

TL;DR

This paper combines computational and theoretical methods to analyze lattice structures related to moduli spaces of K3 surfaces, extending known results to new cases such as d=52.

Contribution

It introduces a computational approach to lattice problems and completes the classification of the moduli space of K3 surfaces for d=52, filling a gap in previous research.

Findings

Solved a combinatorial lattice problem in E8.

Proved the moduli space of K3 surfaces of degree 2d is of general type for d=52.

Extended previous classifications to include the case d=52.

Abstract

We show how to solve computationally a combinatorial problem about the possible number of roots orthogonal to a vector of given length in $E_8$. We show that the moduli space of K3 surfaces with polarisation of degree 2d is also of general type for d=52. This case was omitted from the earlier work of Gritsenko, Hulek and the second author. We also apply this method to some related problems. In Appendix A, V. Gritsenko shows how to arrive at the case d=52 and some others directly.