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 Answer 1


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.)

  • $\begingroup$ 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, 2021 at 19:22
  • $\begingroup$ This is just a standard notation. $\endgroup$
    – abx
    Feb 3, 2021 at 19:36
  • 1
    $\begingroup$ @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 $\endgroup$
    – Samuel
    Feb 4, 2021 at 0:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.