10
$\begingroup$

Let $A$ be an abelian variety over $\mathbb{C}$ and let $X_m$ the subset of nontrivial $m$-torsion points on $A$. Can we realize $X_m$ as the zero locus of a global section of a suitable vector bundle $E$ of rank $\dim(A)$ on $A$?

For $\dim(A)=1$ the answer is trivially yes and for $\dim(A)=2$ this should be doable via the Serre construction. What about higher dimensions?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
15
$\begingroup$

The crucial case is $m=1$: if you have a vector bundle $E$ on $A$ of rank $\dim(A)$ and a section $s$ of $E$ whose zero locus is $\{0\} $, pulling back $(E,s)$ by multiplication by $m$ gives the general case. This question has been studied by O. Debarre, The diagonal property for abelian varieties, Contemporary Mathematics 465, AMS (2008), p. 45-50. The answer is positive for Jacobians, but not for general abelian varieties.

Improve this answer
$\endgroup$
1
  • $\begingroup$ This gives one direction but not the other. Is it true / simple to see that if $A$ is an ab var on which $X_0 = \{0\}$ cannot be realized as a zero locus, then no $X_m$ can? $\endgroup$
    – Kevin Casto
    Commented Mar 12, 2023 at 8:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .