Modular forms on the moduli space of polarised K3 surfaces

TL;DR

This paper analyzes the moduli space of polarised K3 surfaces, computing relations among divisors, describing the Picard group, and determining the Kodaira dimension, with new results for specific degrees.

Contribution

It provides a detailed computation of divisor relations and the Picard group for moduli spaces of polarised K3 surfaces, including new results on their Kodaira dimension.

Findings

Computed all relations between Noether-Lefschetz divisors for degrees up to 50.

Provided a concrete description of the rational Picard group of the moduli space.

Confirmed or established the Kodaira dimension for various degrees.

Abstract

We study the moduli space F_{2d} of polarised K3 surfaces of degree 2d. We compute all relations between Noether-Lefschetz divisors on these moduli spaces for d up to around 50. This leads to a very concrete description of the rational Picard group of F_{2d}. We show how to determine the coefficients of boundary components of relations in the rational Picard group, giving relations on a (toroidal) compactification of F_{2d}. We draw conclusions from this about the Kodaira dimension of F_{2d}, in many cases confirming earlier results by Gritsenko, Hulek and Sankaran, and in two cases giving a new result. This is an abridged version of the PhD thesis by the same author.