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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Enumerating pencils with moving ramification on curves

Author: Carl Lian
Journal: J. Algebraic Geom. 32 (2023), 143-182
Published electronically: December 28, 2021
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Abstract | References | Additional Information

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 $E\to \mathbb {P}^1$ 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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Additional Information

Carl Lian
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
MR Author ID: 1096109

Received by editor(s): October 6, 2020
Received by editor(s) in revised form: November 10, 2020
Published electronically: December 28, 2021
Additional Notes: This project was undertaken with the support of an NSF Graduate Research Fellowship
Article copyright: © Copyright 2021 University Press, Inc.