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# Exceptional Covers of Surfaces

Published:2010-05-11
Printed: Sep 2010
• Jeffrey D. Achter,
Department of Mathematics, Colorado State University, Fort Collins, CO, USA
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## Abstract

Consider a finite morphism $f: X \rightarrow Y$ of smooth, projective varieties over a finite field $\mathbf{F}$. Suppose $X$ is the vanishing locus in $\mathbf{P}^N$ 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 $|{\mathbf{F}}|>C$, then $f(\mathbf{F}): X(\mathbf{F}) \rightarrow Y(\mathbf{F})$ is injective if and only if it is surjective.
 MSC Classifications: 11G25 - Varieties over finite and local fields [See also 14G15, 14G20]