On self-correspondences on curves

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 ${\pi }_1$ and ${\pi }_2$ from $D$ to $C$. A subset $S$ of $C$ is complete if ${\pi }_1^{-1}(S)={\pi }_2^{-1}(S)$. 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 $\mathrm{deg}<!--FUNCTION\; APPLICATION-->{\pi }_1=\mathrm{deg}<!--FUNCTION\; APPLICATION-->{\pi }_2$ 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.