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

Abstract: We consider the quot scheme $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$ of locally free quotients of $\mathcal F^r:= \bigoplus ^{ r} \mathcal O_{\mathbb P^1 }$ 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 $\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k}$, called Hilb-support that allows us to define a natural surjective morphism of schemes $\xi :\mathrm{Quot}^d_{\mathcal F^r/ \mathbb P^1/ k} \to \mathrm{Hilb}^d_{\mathcal O_{\mathbb P^1}} $ associating to each module its Hilb-support and study the fibres of $\xi$ over each $k$-point $Z$ of $\mathrm{Hilb}^d_{\mathcal O_{\mathbb P^1}}$. If $Z=Y_1+\dots+Y_n$, with $Y_j=t_jR_j$, where $R_1, \dots, R_n$ are distinct points, the fibre of $\xi$ over $Z$ is isomorphic to $\mathrm{Quot}^{t_1}_{\mathcal F\otimes \mathcal O_{Y_1}/ Y_1/ k}\times\dots \times \mathrm{Quot}^{t_n}_{\mathcal F\otimes \mathcal O_{Y_n}/ Y_n/ k}$. We then study the Quot scheme $\mathrm{Quot}^{t}_{\mathcal F^r\otimes \mathcal O_{Y}/ Y/ k}$ with $Y=tR$. For $t=1$, $\mathrm{Quot}^{t}_{\mathcal F^r\otimes \mathcal O_{Y}/ Y/ k}$ is isomorphic to $\mathbb P^{r-1}$, while for $t\geq 2$ 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.