# Flat familiy of coherent sheaves over a scheme

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?

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.)
• 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? Feb 3, 2021 at 19:22
• @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 Feb 4, 2021 at 0:07