Generic points of the Quot scheme

Question

Given a projective scheme X, let we consider the Quot scheme $ Quot (X, N, P) $ of the coherent sheaves quotient of $\mathcal {O}_X^N $ and with given Hilbert polinomial P. Of course these sheaves represent the closed points in the Quot. What is an open point in the Quot? I am looking for a geometric interpretation, if there is one.

Answer 1

By definition Quot represents the functor sending $T$ to the set of isomorphism classes of quotients of $\mathcal{O}_{X\times T}^N$, flat over $T$ and with Hilbert polynomial $P$ on the fibers.

If $p$ is any point of Quot, then you can see it as a morphism from $Spec (k(p))$ to Quot, and as such it corresponds to a quotient of $\mathcal{O}_{X_{k(p)}}^N$ on $X_{k(p)}$ with Hilbert polynomial $P$. If $p$ is closed and you are over an algebraically closed base field $k$, then of course $k(p)=k$ and you just get sheaves on $X$ like you said. If $p$ is not closed, instead of shaves on $X$ you get sheaves on the base-change $X_{k(p)}$.

(or, if you want something even more geometric, you can take the closure $Y=\overline{\{p\}}$ of $p$ in Quot, and the immersion from $Y$ to Quot will correspond to a quotient of $\mathcal{O}_{X\times Y}^N$ on $X\times Y$, flat over $Y$ and with Hilbert polynomial $P$ on the fibers)