On self-correspondences on curves

Bellaïche, Joël

doi:10.2140/ant.2023.17.1867

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On self-correspondences on curves

Joël Bellaïche

Vol. 17 (2023), No. 11, 1867–1899

DOI: 10.2140/ant.2023.17.1867

Abstract

We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve $C$ over an algebraically closed field is the data of another curve $D$ and two nonconstant separable morphisms $π_{1}$ and $π_{2}$ from $D$ to $C$ . A subset $S$ of $C$ is complete if $π_{1}^{- 1} (S) = π_{2}^{- 1} (S)$ . We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which $C$ is a union of finite complete sets. The latter ones are called finitary, and happen only when $\deg π_{1} = \deg π_{2}$ and have a trivial dynamics. For a nonfinitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets.

Keywords

algebraic curve, self-correspondence, algebraic dynamics

Mathematical Subject Classification

Primary: 14A10, 37E99

Milestones

Received: 5 March 2021

Revised: 22 April 2022

Accepted: 18 August 2022

Published: 3 October 2023

Authors

Joël Bellaïche
Mathematics Department Brandeis University Waltham, MA United States

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Vol. 17, No. 11, 2023