Enumerating pencils with moving ramification on curves

TL;DR

This paper develops formulas for counting branched covers of the projective line from genus 1 curves with moving ramification points, revealing invariance properties and generalizing previous results in enumerative geometry.

Contribution

It provides new formulas for enumerating covers with moving ramification on elliptic curves, utilizing limit linear series and inclusion-exclusion techniques.

Findings

Derived a simple weighted count formula for pencils on elliptic curves.

Established formulas for maps with moving ramification conditions.

Proved invariance of these counts under a specific involution.

Abstract

We consider the general problem of enumerating branched covers of the projective line from a fixed general curve subject to ramification conditions at possibly moving points. Our main computations are in genus 1; the theory of limit linear series allows one to reduce to this case. We first obtain a simple formula for a weighted count of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce, using an inclusion-exclusion procedure, formulas for the numbers of maps E->P^1 with moving ramification conditions. A striking consequence is the invariance of these counts under a certain involution. Our results generalize work of Harris, Logan, Osserman, and Farkas-Moschetti-Naranjo-Pirola.