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On self-correspondences on curves

Joël Bellaïche

Vol. 17 (2023), No. 11, 1867–1899
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 π1 and π2 from D to C. A subset S of C is complete if π11(S) = π21(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 π1 = deg π2 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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