The Picard group of moduli spaces of K3 surfaces and their compactifications

The study of Picard group of moduli spaces is an important question in moduli theory. Since from Mumford's famous paper “Picard groups of moduli problems”, algebraic geometers have investigated the Picard group of moduli space of smooth curves and its compactifications. It has been shown by Harer and Arbarello-Cornalba that $M_g$ has Picard number one while its Deligne-Mumford compactification $\overline{M}_g$ has Picard number $2+[g/2]$. In this talk, I will survey the recent progress for moduli spaces of projective K3 surfaces and their compactifications. We will give a complete description of their Picard groups.