# Exceptional covers of surfaces

Jeff Achter

Arxiv ID: 0707.2612•Last updated: 2/27/2020

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.

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