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≠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 Pic_Q(K_g) using the Noether–Lefschetz (NL) theory. This verifies the NL conjecture on the moduli of K3 surfaces in these cases