Harvard-MIT Algebraic Geometry Seminar: A degeneration of the Hilbert scheme

Grothendieck's Hilbert scheme is a compact parameter space for subschemes of a projective scheme X. It is one of the basic moduli spaces in algebraic geometry, in the sense that it is the starting point for the construction of many others. One simple question about the Hilbert scheme is the following: as X undergoes a nice degeneration, what is the right way to degenerate the Hilbert scheme of X along with it? One possible answer, proposed in the PhD thesis of Kennedy-Hunt, comes from an object called the logarithmic Hilbert scheme. I will give an introduction to this circle of these ideas, explain the basic geometric properties of the logarithmic Hilbert scheme, and sketch connections with certain moduli spaces of higher dimensional varieties. The talk reports on work of Kennedy-Hunt, joint work with Kennedy-Hunt, and joint work with Maulik.