Torsion points on theta divisors and semihomogeneous vector bundles

TL;DR

This paper extends Kempf's result on 2-torsion points on theta divisors to n-torsion points using semihomogeneous vector bundles, providing a sharp upper bound and confirming a conjecture about products of elliptic curves.

Contribution

It introduces a generalization of torsion point results on theta divisors to arbitrary n, utilizing semihomogeneous vector bundles, and proves a conjecture regarding the maximum number of n-torsion points.

Findings

Established a sharp upper bound for n-torsion points on theta divisors.

Proved that the maximum is achieved only by products of elliptic curves.

Confirmed a conjecture of Auffarth, Pirola, and Salvati Manni.

Abstract

We generalize to $n$-torsion a result of Kempf's describing $2$-torsion points lying on a theta divisor. This is accomplished by means of certain semihomogeneous vector bundles introduced and studied by Mukai and Oprea. As an application, we prove a sharp upper bound for the number of $n$-torsion points on a theta divisor, and show that this is achieved only in the case of products of elliptic curves, settling in the affirmative a conjecture of Auffarth, Pirola and Salvati Manni.