素数次的 Hurwitz 存在性问题
The Hurwitz existence problem in prime degree
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中文总结 AI 辅助
本文证明素数度下球面上所有相容分支数据均可由连通分支覆盖实现，并由此推出 Hurwitz 势的完全支撑及相关 Gromov--Witten 不变量的正性。
中文摘要 AI 辅助
设 $p$ 为素数。我们证明，球面上每个度为 $p$ 的相容分支数据都能由连通分支覆盖实现。三点情形在剩余特征 $p$ 下构造。Henrio 的矩定理提供了不同点的矩解，我们由此为每个指定的划分构造一个特殊的本原尾；第二次应用为正向源亏格所需的新尾奠定了基础。这些尾通过对数形变数据连接，并嵌入 $S_p$ 的一个包含阶为 $p$ 的公共正则子群的子群中。Wewers 的提升定理在特征零下产生一个三点 Galois 覆盖。对点稳定子取商后，度为 $p$，且具有指定的三个分歧轮廓。Edmonds--Kulkarni--Stong 的融合与实现结果随后给出任意多个分支值的断言。作为推论，连通的素数次 Hurwitz 势在 Riemann--Hurwitz 轨迹上具有完全支撑，每个相应的连通相对 Gromov--Witten 不变量（对 $\mathbf P^1$）非零，具有偶数完备循环阶（至多 $p$）的二相对点不连通扇区在维数约束下严格为正，且连通换位扇区严格为正。
英文摘要
Let $p$ be a prime. We prove that every compatible branch datum of degree $p$ over the sphere is realizable by a connected branched cover. The three-point case is constructed in residue characteristic $p$. Henrio's moment theorem supplies the distinct-point moment solutions from which we construct a special primitive tail for each prescribed partition; a second application underlies the new tail required by a positive source genus. These tails are joined by a logarithmic deformation datum and embedded in one subgroup of $S_p$ containing a common regular subgroup of order $p$. Wewers's lifting theorem produces a three-point Galois cover in characteristic zero. The quotient by a point stabilizer has degree $p$ and the prescribed three ramification profiles. The fusion and realization results of Edmonds--Kulkarni--Stong then give the assertion for an arbitrary number of branch values. As consequences, the connected prime-degree Hurwitz potential has full support on the Riemann--Hurwitz locus, every corresponding connected relative Gromov--Witten invariant of $\mathbf P^1$ is nonzero, the two-relative-point disconnected sector with even completed-cycle orders at most $p$ is strictly positive subject to the dimension constraint, and the connected transposition sector is strictly positive.
