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We discuss the Picard group of the moduli space   ${\mathcal {K}}_g$   of quasi-polarized K3 surfaces of genus   $g\leq 12$   and   $g\neq 11$  . 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 for the Picard group   ${\rm Pic}_{\mathbb {Q}}({\mathcal {K}}_g)$   using the Noether–Lefschetz (NL) theory. This verifies the NL conjecture on the moduli of K3 surfaces in these cases.

Picard Groups on Moduli of K3 Surfaces with Mukai Models