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 lying 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.

中文翻译：

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θ 除数和半齐次向量丛上的扭点

我们概括为 $n$- 扭转 Kempf 描述的结果 $2$- 位于 theta 除数上的扭转点。这是通过 Mukai 和 Oprea 引入和研究的某些半齐次向量丛来实现的。作为一个应用，我们证明了数量的急剧上限$n$- 扭转点位于 theta 除数上，并表明这仅在椭圆曲线乘积的情况下才能实现，从而肯定了 Auffarth、Pirola 和 Salvati Manni 的猜想。