Theta对偶性与超椭圆曲线循环覆盖的Prym-Torelli定理
Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves
中文总结 AI 辅助
本文通过theta对偶性重构方法，研究超椭圆曲线奇素数度循环覆盖的Prym映射的通用单射性，识别出额外剩余轨迹，并在不同度下给出单射性条件。
中文摘要 AI 辅助
设$f:\tilde{C}\to C$是亏格$g\geq 2$的超椭圆曲线的一个奇素数度$d$的平展循环覆盖。我们重新审视Naranjo-Ortega-Pirola-Spelta关于相关Prym映射的通用单射性的theta对偶性重构论证，并识别出该论证中出现的线束的固定因子所产生的额外剩余轨迹。对于$d\geq5$，该轨迹不影响重构，在数值假设$(d-1)(g-1)\geq 7$下给出单射性。对于$d=3$，几何由超椭圆对合提升在$\tilde{C}$的商上的三次铅笔所支配；这导致在亏格5时给出通用度2，在$g\geq6$时给出单射性。
英文摘要
Let $f:\tilde{C}\to C$ be an étale cyclic cover of odd prime degree $d$ of a hyperelliptic curve of genus $g\geq 2$. We revisit the theta-duality reconstruction argument for the generic injectivity of the associated Prym map by Naranjo-Ortega-Pirola-Spelta and identify an additional residual locus arising from the fixed divisor of the line bundles occurring in that argument. For $d\ge5$, this locus does not affect reconstruction, giving injectivity under the numerical assumption $(d-1)(g-1)\geq 7$. For $d=3$, the geometry is governed instead by trigonal pencils on the quotient of $\tilde{C}$ by a lift of the hyperelliptic involution; this yields generic degree $2$ in genus $5$ and injectivity for $g\ge6$.
