Exceptional covers of surfaces

TL;DR

This paper establishes a condition under which a finite morphism between smooth projective varieties over a finite field is bijective on rational points, based on the size of the field and geometric properties of the varieties.

Contribution

It proves a new criterion linking injectivity and surjectivity of morphisms over finite fields, depending only on geometric parameters and the field size.

Findings

Existence of a constant C depending on geometric data

Injectivity and surjectivity are equivalent for large enough finite fields

The result applies to varieties defined by low-degree forms

Abstract

Consider a finite morphism f:X -> Y of smooth projective varieties over a finite field k. Suppose X is the vanishing locus in projective N-space of at most r forms of degree at most d. We show there is a constant C, depending only on N, r, d and deg(f) such that if #k > C, then f(k):X(k) -> Y(k) is injective if and only if it's surjective.