I'm studying Quot schemes, that I denote with $Quot_{N,X,P}$, with $N \in \mathbb{Z}$, $X \subset \mathbb{P}^d$ and $P \in \mathbb{Q}[t]$. So, I'm looking for explicit examples of Quot schemes. Could you give me as much as you can explicit examples of Quot schemes? On the net I found several papers about the construction of Quot schemes, but I can't find explicit examples in order to have a true understanding of these objects. For instance, how can I think of the Grassmannian as $Quot_{N,X,P}$ scheme? Are there different ways to do this? I'm sorry if this question isn't specific enough. Thanks in advance.

$\begingroup$ Would some of the links provided by a Google search for "explicit examples of Quot schemes" be useful? (It's not my specialization). $\endgroup$ – Włodzimierz Holsztyński Nov 16 '13 at 16:19

$\begingroup$ @WlodzimierzHolsztynski I'm sorry, but unfortunately I can't find them! $\endgroup$ – Oscar1778 Nov 16 '13 at 16:40

$\begingroup$ msp.org/pjm/2007/2312/pjmv231n2p07s.pdf: "...explicit...of quot schemes..." by Gustavsen at al $\endgroup$ – Włodzimierz Holsztyński Nov 16 '13 at 17:06

$\begingroup$ algant.eu/documents/theses/habibi.pdf: "Hilbert and Quot Schemes" by S.Habibi $\endgroup$ – Włodzimierz Holsztyński Nov 16 '13 at 17:10
$\begingroup$
$\endgroup$
You'll find the answer to some of your questions in N. Nitsure, Construction of Hilbert and Quot Schemes, arXiv:math/0504590.

1$\begingroup$ Right, that's what a Google search for "explicit examples of Quot schemes" provides (and more). $\endgroup$ – Włodzimierz Holsztyński Nov 16 '13 at 17:14