Title: Reverse Hurwitz counts of genus 1 curves Show Chinese title

Title: 1类曲线的逆Hurwitz计数

Abstract: In this paper, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target -- $\mathbb P^1$ with a list of $n$ points $q_1,\dots,q_n\in \mathbb P^1$ -- and partitions $\sigma_1,\dots,\sigma_n$ of $d$, how many degree $d$ covers $C\to\mathbb P^1$ are there with specified ramification $\sigma_i$ over $q_i$? We ask: for a generic source -- an $r$-pointed curve $(C,p_1,\dots,p_r)$ of genus $1$ -- and partitions $\mu, \sigma_1,\dots,\sigma_n$ of $d$ with $\ell(\mu)=r$, how many degree $d$ covers $C\to\mathbb P^1$ are there with ramification profile $\mu$ over $0$ corresponding to a fiber $\{p_1,\dots,p_r\}$ and elsewhere ramification profiles $\sigma_1,\dots,\sigma_n$? While the enumerative invariants we study bear a similarity to generalized Tevelev degrees, they are more difficult to express in closed form in general. Nonetheless, we establish key results: after proving a closed form result in the case where the only non-simple unmarked ramification profiles $\sigma_1$ and $\sigma_2$ are ``even'' (consisting of $2,\dots,2$), we go on to establish recursive formulas to compute invariants where each unmarked ramification profile is of the form $(x,1,\dots,1)$. A special case asks: given a generic $d$-pointed genus $1$ curve $(E,p_1,\dots,p_d)$, how many degree $d$ covers $(E,p_1,\dots,p_d)\to(\mathbb P^1,0)$ are there with $d-2$ unspecified points of $E$ having ramification index $3$? We show that the answer is an explicit quartic in $d$. Show Chinese abstract

Abstract: 在本文中，我们研究了一个在某种意义上是Hurwitz计数问题的反向问题。 Hurwitz问题询问：对于一个一般的目标--$\mathbb P^1$，带有列表中的$n$个点$q_1,\dots,q_n\in \mathbb P^1$--以及$\sigma_1,\dots,\sigma_n$的$d$的划分，有多少个次数为$d$的覆盖$C\to\mathbb P^1$在$q_i$上有指定的分歧$\sigma_i$？ 我们问：对于一个一般的源——一个 genus$1$的$r$点曲线$(C,p_1,\dots,p_r)$——以及 partitions$\mu, \sigma_1,\dots,\sigma_n$of$d$with$\ell(\mu)=r$，有多少个 degree$d$的 covers$C\to\mathbb P^1$在对应于一个 fiber$\{p_1,\dots,p_r\}$的$0$上具有分歧轮廓$\mu$，而在其他地方具有分歧轮廓$\sigma_1,\dots,\sigma_n$? 虽然我们研究的枚举不变量与广义Tevelev度相似，但通常更难以用闭合形式表达。尽管如此，我们建立了关键结果：在仅存在非简单未标记分支剖面$\sigma_1$和$\sigma_2$为“偶数”（由$2,\dots,2$组成）的情况下，我们证明了闭合形式的结果，然后继续建立递归公式来计算每个未标记分支剖面均为$(x,1,\dots,1)$形式的不变量。 一个特殊情况是：给定一个一般的$d$点的亏格$1$曲线$(E,p_1,\dots,p_d)$，有多少个次数为$d$的覆盖$(E,p_1,\dots,p_d)\to(\mathbb P^1,0)$，其中$d-2$个未指定的点的$E$具有分歧指数$3$？ 我们证明答案是$d$的一个显式四次方程。