Matrix points on varieties and punctual Hilbert (and Quot) schemes

Abstract: Moduli spaces often have interesting enumerative properties. The goal of this talk is to introduce some enumerative results on solutions of matrix equations and zero-dimensional sheaves over singular curves. To motivate them, I first discuss several moduli spaces in general, which I put onto the "unframed" side and the "framed" side. The unframed side includes the commuting variety AB=BA of n x n matrices, the variety of commuting matrices satisfying polynomial equations (the titular "matrix points on varieties"), and the moduli stack of zero-dimensional coherent sheaves on a variety. The framed side includes the Hilbert scheme of points on a variety, or more generally, the Quot scheme of zero-dimensional quotients of a vector bundle on a variety. The enumerative properties to be considered are point counts over finite fields and the motive in the Grothendieck ring of varieties, which essentially keep track of the combinatorial data of a stratification of the space in question. I will explain some general connections within and between the two sides, and known results for smooth curves and smooth surfaces. Finally, I will discuss recent results on singular curves. This talk is based on joint work with Ruofan Jiang.

In the pre-seminar, I plan to talk about a super fun combinatorial construction, which we call “spiral shifting operators”, used in the proof of one of our results.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: Description: Research/departmental seminar

Seminar usually meets in-person. For online editions, the Zoom link is shared via mailing list.