Intersection of a correspondence with a graph of Frobenius

TL;DR

This paper provides a concise geometric proof of Hrushovski's theorem, demonstrating that the intersection of a correspondence with a sufficiently large power of Frobenius's graph is always non-empty.

Contribution

It offers a new geometric proof of a key theorem relating to Frobenius and correspondences, simplifying previous approaches.

Findings

Proves the non-emptiness of the intersection for large Frobenius powers.

Introduces a geometric perspective to a previously algebraic proof.

Simplifies understanding of Frobenius correspondences in algebraic geometry.

Abstract

The goal of this note is to give a short geometric proof of a theorem of Hrushovski asserting that an intersection of a correspondence with a graph of a sufficiently large power of Frobenius is non-empty.