The Picard group of a variety, i.e. the group of line bundles that live over it, is an interesting and much studied invariant in algebraic geometry. In particular, the Picard groups of moduli spaces of objects (e.g. smooth curves of fixed genus, quasi-polarized K3 surfaces of fixed degree) have been the subject of a large amount of research in the past thirty years.
In this talk, I will explain how techniques of equivariant intersection theory, applied to certain moduli spaces of complete intersections, can shed some light on two specific questions:
(1) Suppose that the base field has positive characteristic. Is the Picard group of the moduli stack of smooth curves of fixed genus freely generated by the Hodge line bundle?
(2) Do the Noether-Lefschetz elliptic divisors freely generate the integral Picard group of moduli stack of polarized K3 surfaces?
Time permitting, I will show how the same methods can be used in order to compute the classes of some geometrically meaningful divisors on these moduli spaces in terms of the generators of the Picard groups.