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Published online by Cambridge University Press: 20 November 2018
Consider a finite morphism $f\,:\,X\,\to \,Y$ of smooth, projective varieties over a finite field
$\mathbb{F}$. Suppose
$X$ is the vanishing locus in
${{\mathbb{P}}^{N}}$ of
$r$ forms of degree at most
$d$. We show that there is a constant
$C$ depending only on
$(N,\,r,\,d)$ and
$\deg (f)$ such that if
$\left| \mathbb{F} \right|\,>\,C$, then
$f\,(\mathbb{F})\,:\,X(\mathbb{F})\,\to Y(\mathbb{F})$ is injective if and only if it is surjective.