Picard Groups on Moduli of K3 Surfaces with Mukai Models

We discuss the Picard group of the moduli space K-g of quasi-polarized K3 surfaces of genus g <= 12 and g not equal 11. In this range, K-g is unirational, and a general element in 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 for the Picard group PicQ(K-g) using the Noether-Lefschetz (NL) theory. This verifies the NL conjecture on the moduli of K3 surfaces in these cases.