Pl\"ucker degrees of Quot schemes

TL;DR

This paper investigates the Plücker degree of Quot schemes on smooth projective schemes, providing a decomposition of classes in the Chow ring to compute leading terms and extending classical results to higher dimensions.

Contribution

It introduces a new decomposition method for classes in the Chow ring to compute Plücker degrees and generalizes classical Schubert results to higher-dimensional cases.

Findings

Decomposition of classes in the Chow ring for Quot schemes.

Explicit computation of the leading term of the Plücker degree.

Higher-dimensional analogue of classical Schubert results.

Abstract

We study the Pl\"{u}cker degree of the main component of the Quot scheme of length $l$ quotients of a locally free sheaf on a smooth projective scheme $\mathrm{S}$ of dimension $d\geqslant 1$. This degree is determined by classes in the Chow ring of the symmetric product $\mathrm{S}^{(l)}$, which are given by the pushforward of the powers of $c_{1}(\mathcal{O}^{[l]})$ with respect to the canonical morphism from the Quot scheme to $\mathrm{S}^{(l)}$. We describe a decomposition of these classes, allowing us to compute the (in a certain sense) leading term of the Pl\"{u}cker degree. We also obtain a higher-dimensional analogue of a classical result of Schubert.