1
$\begingroup$

This is the definition given in Vistoli's paper.

Let $F$ be an integral stack. A rational function of $F$ is a morphism $G \rightarrow A^1_S$ defined on a nonempty open substack $G$ of $F$.

Rational functions can be added and multiplied, and they form a field $K(F)$, the quotient field of $F$.

Then he gives a simple example. A rational function on $[X/G]$ is an invariant rational function on $X$.

I have no trouble understanding this. But it's hard for me to think about the general case. I think a rational function is defined on some atlas $U$ over some open substack of $F$ and satisfies certain gluing conditions.

Can someone give some other examples which are not quotient stacks?

Also in his proof of the proposition 1.17, I don't understand why $k(F)=k(M)$.

enter image description here

Improve this question
$\endgroup$
3
  • $\begingroup$ You want examples of fields of rational functions? Do you understand the concept for varieties? For example do you know what the field of rational functions on affine $n$-space is? If so, can you make the question a bit clearer? $\endgroup$
    – znt
    Commented Sep 17, 2016 at 20:40
  • $\begingroup$ @znt I want an example which is not a quotient stack and definitely not a scheme. $\endgroup$
    – WWK
    Commented Sep 17, 2016 at 21:04
  • 3
    $\begingroup$ Any morphism from a stack to $\mathbb A_1$ factors through the coarse moduli space, so you can just think of them as rational functions on $M$. $\endgroup$
    – François
    Commented Sep 18, 2016 at 4:21

0

Reset to default

You must log in to answer this question.