5

I have been learning about fibered categories and stacks from Vistoli's notes. One of the main results in the notes is the statement that the fibered category of quasicoherent sheaves over a scheme $X$ is a stack in the fpqc topology on the category of $X$-schemes. I can appreciate that this is a surprising result, as quasicoherent sheaves are a priori constructed as a Zariski stack, and the fpqc topology is strictly finer than the Zariski topology. Incidentally, I also think the proof presented in the notes is good practice with the concepts he introduces.

I am wondering about applications of this result, as they are not really mentioned in the notes. I am not familiar with descent theory outside of what is discussed in Vistoli, so I'm partly asking this to get a feel for the topic - a sort of "what's next?" The question: What are some examples of interesting results in which in some way use the fact that the fibered category of quasicoherent sheaves over a scheme is a stack in the fpqc topology?

Alex K
  • 531
  • 1
    This allows you to prove descent for the moduli space of polarized varieties, for example the moduli space of polarized K3 surfaces. – Pol van Hoften May 05 '20 at 09:01

1 Answers1

3

I don't know of any immediate applications, but this is a key result in showing you can descend curves (with the appropriate definition, see references) of genus $g \neq 1$. This is mentioned in example 4.39 of Vistoli's notes. A thorough investigation of this is given in this master's thesis. One wants descent for projective morphisms, which only works if they come with an ample line bundle with descent data - this is where we need the result about quasicoherent sheaves.

As Vistoli notes, this means that $\mathcal{M}_g$, the moduli space of genus $g$ smooth curves, is a stack. Deligne and Mumford study it in this paper, and use the fact that it's a DM-stack to prove some cool results about the geometry of the moduli space of smooth curves. I'm told that this really uses the fact that $\mathcal{M}_g$ is a stack, and hence better behaved than the coarse moduli space. For example, the coarse moduli space is not smooth, whereas $\mathcal{M}_g$ is a smooth DM-stack.

Finally, this is not an application, but the proof that QCoh is a stack is a prototypical example of fpqc descent, from which you can deduce many other cases such as descent for quasicoherent $\mathcal{O}_X$-algebras or (quasi-)affine morphisms.

ggg
  • 1,383