Plücker degrees of Quot schemes

Abstract

We study the degree of the Plücker embedding $\varpi$ 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 $\mathrm{deg} \ \varpi$. We also obtain a higher-dimensional analogue of a classical result of Schubert.

Theorems & Definitions (19)

• Theorem I
• Theorem II
• Theorem 1.1: Grothendieck
• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 3.1
• proof