0
$\begingroup$

I am new to study of abelian varieties. But I need it in my work. Let $X$ be a ppav, say a Jacobian of a genus 2 curve. Let $L$ be a very ample line bundle on $X$.

The set $K(L)=\{x\in X : T_x^* L\simeq L\}$ is a finite set since $L$ is ample.

1) By theorem of square, if $x\in K(L)$, all multiples are in $K(L)$ also. How is this possible. Does it mean $K(L)$ consist of only torsion elements?

2) Does $K(L)$ act on $|L|$? Take curve $C\in |L|$ then $C-x\in |L|$ right? What can we say about this action?

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ The answers to both questions are "yes". $\endgroup$
    – Jason Starr
    Commented Sep 24, 2016 at 15:00
  • $\begingroup$ @Jason Starr, thank you. What is the orbit of a curve $C$? $\endgroup$
    – user52991
    Commented Sep 24, 2016 at 15:06
  • $\begingroup$ What do you mean by "what is the orbit of a curve $C$?" For a generic choice of divisor in the linear system of $|L|$, $K(L)$ will act freely with trivial stabilizer and with orbit of cardinality $\# K(L)$. Is that what you are asking? There can be divisors with nontrivial stabilizer, e.g., the flex lines to a fixed plane cubic $X$ in $|L|=\mathbb{P}^2$. $\endgroup$
    – Jason Starr
    Commented Sep 24, 2016 at 15:19
  • $\begingroup$ @Jason Starr, thank you, that is what I wanted to know. Is there some reference to this? $\endgroup$
    – user52991
    Commented Sep 24, 2016 at 15:21
  • $\begingroup$ "Is there some reference to this?" Example 9.3, p. 141, of Igor Dolgachev, "Lectures on Invariant Theory". $\endgroup$
    – Jason Starr
    Commented Sep 24, 2016 at 15:26

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .