# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.09087