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Punctual Quot scheme on cusp via Gröbner stratification

Yifeng Huang (UBC)
Tuesday, April 18, 2023 - 1:45pm
PDL C-38
Title pre-talk: Hilbert schemes of points on singular curves
Abstract pre-talk: I will recall the definition and basic properties of Hilbert schemes of points on a variety, and introduce some beautiful theorems and conjectures about Hilbert schemes of points on singular curves in the last decade. Through concrete examples, I will explain how Gröbner basis theory and Serre duality can be applied here. 
Title main talk: Punctual Quot scheme on cusp via Gröbner stratification
Abstract main talk: (Joint with Ruofan Jiang) We prove a rationality result for a zeta function for the Quot scheme of points on the cusp singularity x^2=y^3, extending a phenomenon that is known for the Hilbert scheme of points on singular curves. The Quot scheme in question parametrizes quotient sheaves of O_X^d of length n supported at p, where p is a cusp singularity on a curve X; the Hilbert scheme is the special case d=1. Our method is based on a stratification given by Gröbner bases for power series ring (a.k.a. standard bases). The essential combinatorial ingredient behind the rationality is a family of “spiral shifting” operators that act on the index set of the strata, which group the (infinitely many) strata into finitely many ``orbits’’. Our method permits explicit computations for low d, which suggest several geometric conjectures that potentially hold for other curve singularities. 

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