Picard groups on moduli of K3 surfaces with Mukai models

Abstract

We discuss the Picard group of moduli space ${\mathcal{K}}_g$ of quasi-polarized K3 surfaces of genus $g\le 12$ and $g\ne 11$. In this range, ${\mathcal{K}}_g$ is unirational and a general element in ${\mathcal{K}}_g$ is a complete intersection with respect to a vector bundle on a homogenous space, by the work of Mukai. In this paper, we find generators of the Picard group $Pi{c}_{\mathbb{Q}}({\mathcal{K}}_g)$ using Noether-Lefschetz theory. This verifies the Noether-Lefschetz conjecture on moduli of K3 surfaces in these cases.

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