On the Quot scheme ${\mathrm{Quot}}_{{\mathcal{O}}_{{\mathbb{P}}^1}^r/{\mathbb{P}}^1/k}^d$

Abstract

We consider the quot scheme ${\mathrm{Quot}}_{{\mathcal{F}}^r/{\mathbb{P}}^1/k}^d$ of locally free quotients of ${\mathcal{F}}^r:={⨁}^r{\mathcal{O}}_{{\mathbb{P}}^1}$ with Hilbert polynomial $p(t)=d$. We prove that it is a smooth variety of dimension $dr$, locally isomorphic to ${\mathbb{A}}^{dr}$. We introduce a new notion of support for modules in ${\mathrm{Quot}}_{{\mathcal{F}}^r/{\mathbb{P}}^1/k}^d$, called Hilb-support that allows us to define a natural surjective morphism of schemes $\xi :{\mathrm{Quot}}_{{\mathcal{F}}^r/{\mathbb{P}}^1/k}^d\to {\mathrm{Hilb}}_{{\mathcal{O}}_{{\mathbb{P}}^1}}^d$ associating to each module its Hilb-support and study the fibres of $\xi$ over each $k$-point $Z$ of ${\mathrm{Hilb}}_{{\mathcal{O}}_{{\mathbb{P}}^1}}^d$. If $Z=Y_1+\cdots +Y_n$, with $Y_j=t_j{R}_j$, where $R_1,\dots ,R_n$ are distinct points, the fibre of $\xi$ over $Z$ is isomorphic to ${\mathrm{Quot}}_{\mathcal{F}\otimes {\mathcal{O}}_{Y_1}/Y_1/k}^{t_1}\times \cdots \times {\mathrm{Quot}}_{\mathcal{F}\otimes {\mathcal{O}}_{Y_n}/Y_n/k}^{t_n}$. We then study the Quot scheme ${\mathrm{Quot}}_{{\mathcal{F}}^r\otimes {\mathcal{O}}_Y/Y/k}^t$ with $Y=tR$. For $t=1$, ${\mathrm{Quot}}_{{\mathcal{F}}^r\otimes {\mathcal{O}}_Y/Y/k}^t$ is isomorphic to ${\mathbb{P}}^{r-1}$, while for $t\ge 2$ we prove that it is formed by a main irreducible, reduced and singular component of dimension $t(r-1)$ and by some embedded component of lower dimension.

11 pages, preliminary version, comments are welcome!