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I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully understand a definition. This could be a very basic question in this topic but it's the first time I'm approaching to it and I would like to have a clear idea of the argument.

Let $X$ be a connected smooth projective curve over an algebraically closed field $K$, then we have the following

Definition: A family over a scheme $S$ of (semi)-stable vector bundles on $X$ with invariants $(n,d)$ is a coherent sheaf $\mathcal{E}$ over $X\times S$ which is flat over $S$ and such that for each $s\in S$, the sheaf $\mathcal{E}_s$ is a (semi)-stable vector bundle on $X$ with invariants $(n,d)$

First of all I want to clarify some of the notation. The so-called invariants $(n,d)$ are indeed respectively, the rank and the degree of each vector bundle. Moreover, the symbol $\mathcal{E}_s$ is meaning that for each $K$-point $s: Spec(K)\rightarrow S$ we can consider the map $s^*:\mathcal{A}_S\rightarrow\mathcal{A}_{Spec(K)}$ from the the family over $S$ to the family over a point and then define $\mathcal{E}_s:=s^*\mathcal{E}$

Question 1

If from the definition $\mathcal{E}$ is a coherent sheaf over $X\times S$, how can I obtain $\mathcal{E}_s$ to be a vector bundles over $X$ if I'm taking the "family pull-back" from $S$ to $Spec(K)$?

In the definition I wrote the author require that the coherent sheaf $\mathcal{E}$ is flat over $S$, as far as I know this means that taken the projection $\pi_S:X\times S\rightarrow S$ then for all $(x,s)\in X\times S$ the stalk $\mathcal{E}_{(x,s)}$ is a flat $\mathcal{O}_{S,s}$-module (with such a structure given by $\pi_S^\sharp:\mathcal{O}_{S,s}\rightarrow\mathcal{O}_{X\times S, (x,s)}$).

Question 2

Why do we require flatness over $S$ in this special case of family of vector bundles? Is there a very simple explanation?

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1.) $\mathscr{E}_s$ is the restriction of $\mathscr{E}$ to $\mathrm{X}\times\{s\}$.

2.) In algebraic geometry flatness gives the right notion of "family". If you assume $\mathscr{E}$ is flat over some connected $\mathrm{S}$, then the Hilbert polynomial of $\mathscr{E}_s$ is independent of $s$; in particular the degree and rank of $\mathscr{E}_s$ are independent of $s$. (There are many other reasons why flatness is nice, of course, but this is the most obvious one.)

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  • $\begingroup$ Thanks. I also thought that $\mathcal{E}_s$ should be such a restriction but I can't see why from the definition of the author, could you give me a hint? $\endgroup$ – John117 Feb 3 at 19:22
  • $\begingroup$ This is just a standard notation. $\endgroup$ – abx Feb 3 at 19:36
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    $\begingroup$ @John117 the restriction is by the definition the pullback of $\mathscr{E}$ under the map $\mathrm{X}\times\{s\}\rightarrow\mathrm{X}\times\mathrm{S}$. If you read your reference carefully, you will find this exactly the definition of the author. There is nothing to it, your confusion is a purely notational one $\endgroup$ – SWS Feb 4 at 0:07

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