On the motive of the nested Quot scheme of points on a curve

TL;DR

This paper computes the generating function of motives for nested Quot schemes of points on a smooth curve, revealing it factors into a product of motivic zeta functions and is rational.

Contribution

It provides an explicit motivic generating function for nested Quot schemes on a curve, using Bialynicki-Birula decomposition, and shows it factors into motivic zeta functions.

Findings

The generating function is a rational function.

The series factors into a product of shifted motivic zeta functions.

The approach uses Bialynicki-Birula decomposition.

Abstract

Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$, and let $E$ be a locally free sheaf of rank $r$. We compute, for every $d>0$, the generating function of the motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )] \in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is the nested Quot scheme of points, parametrising $0$-dimensional subsequent quotients $E \twoheadrightarrow T_d \twoheadrightarrow \cdots \twoheadrightarrow T_1$ of fixed length $n_i = \chi(T_i)$. The resulting series, obtained by exploiting the Bialynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of $C$. In particular, it is a rational function.