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

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 $[{\mathrm{Quot}}_C(E,\mathbf{n})]\in K_0({\mathrm{Var}}_{\mathbf{k}})$, varying $\mathbf{n}=(0\le n_1\le \cdots \le n_d)$, where ${\mathrm{Quot}}_C(E,\mathbf{n})$ 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.

Fixed typo in the main Theorem, published version to appear in Journal of Algebra, 13 pages

References in corpus (1)

• On the motive of zero-dimensional Quot schemes on a curve

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• The motive of the Hilbert scheme of points in all dimensions
• On quasimap invariants of moduli spaces of Higgs bundles
• The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage
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• Lattices in ${\mathbb{F}}_q[[T]{]}^d$ and spiral shifting operators
• Sur la lissité du schéma Quot ponctuel emboîté