Abstract:
Suppose there are two non-constant separable morphisms from a curve $D$ to a
smooth curve $C$, everything is defined over a field $k$. A self-correspondence on
$C$ is the data consisting of the curve $D$ and these two morphisms. Consider an
algebraic closure $K$ of the field $k$. A self-correspondence can be thought of as
a multi-valued map from $C(K)$ to itself, defined by polynomials with
coefficients in the original field. We will discuss finite subsets of the
curve $C$ which remain fixed under the self-correspondence. Following the paper
by J. Bellaïche, I will explain when there are infinitely many such subsets,
and how many there can be in the opposite case.
