Generalized Tevelev degrees of $\mathbb{P}^1$

TL;DR

This paper extends the enumeration of covers of the projective line by a general curve, incorporating arbitrary ramification profiles and additional incidence conditions, building on intersection theory and limit linear series methods.

Contribution

It generalizes previous counts of covers by including arbitrary ramification profiles and equal image constraints, unifying intersection theory and limit linear series approaches.

Findings

Derived generalized counts for covers with arbitrary ramification.

Unified intersection theory and limit linear series methods.

Extended enumerative formulas to more complex ramification conditions.

Abstract

Let $(C,p_1,\ldots,p_n)$ be a general curve. We consider the problem of enumerating covers of the projective line by $C$ subject to incidence conditions at the marked points. These counts have been obtained by the first named author with Pandharipande and Schmitt via intersection theory on Hurwitz spaces and by the second named author with Farkas via limit linear series. In this paper, we build on these two approaches to generalize these counts to the situation where the covers are constrained to have arbitrary ramification profiles: that is, additional ramification conditions are imposed at the marked points, and some collections of marked points are constrained to have equal image.