240305 Younghan Bae

Moduli spaces of smooth, or stable nodal, curves have been an important subject of algebraic geometry, topology, and mathematical physics. There is a natural way to relate this moduli space with the moduli space of abelian varieties. Over the moduli space of smooth curves M_g, the relative Picard scheme gives a family of principally polarized abelian varieties (p.p.a.v.). Taking this perspective, one can obtain many interesting results on M_g. When curves degenerate into nodal curves, the relative Picard scheme is no longer compact. One can compactly such family and get relative compactified Picard schemes which is no longer a family of p.p.a.v This geometry brings many new geometry and topology. I will try to introduce some new questions on this direction. It will be a gentle overview of a lecture series which I will deliver this block.