Matrix points on varieties and punctual Hilbert (and Quot) schemes
Yifeng Huang (University of British Columbia)
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
Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).
We acknowledge the support of PIMS, NSERC, and SFU.
For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.
We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca
| Organizer: | Katrina Honigs* |
| *contact for this listing |