Integral Picard group of some stacks of polarized K3 surfaces of low degree

TL;DR

This paper computes the integral Picard group of stacks of polarized K3 surfaces with rational double points for degrees 4, 6, and 8, showing it is torsion-free and generated by specific divisors and the Hodge bundle.

Contribution

It provides the first explicit description of the integral Picard group for these stacks, including a basis and relations among divisors, using equivariant geometry techniques.

Findings

Picard group is torsion-free for degrees 4, 6, 8

A basis includes elliptic Noether-Lefschetz divisors and the Hodge bundle

Explicit class expressions for Noether-Lefschetz divisors

Abstract

We compute the integral Picard group of the stack $\mathcal{M}_{2l}$ of polarized K3 surfaces with at most rational double points of degree $2l=4,6,8$. We show that in this range the integral Picard group is torsion-free and that a basis is given by certain elliptic Noether-Lefschetz divisors together with the Hodge line bundle. To achieve this result, we investigate certain stacks of complete intersections and their Picard groups by means of equivariant geometry. In the end we compute an expression of the class of some Noether-Lefschetz divisors, restricted to an open substack of $\mathcal{M}_{2l}$, in terms of the basis mentioned above.