Exceptional Covers of Surfaces

http://dx.doi.org/10.4153/CMB-2010-049-7
Canad. Math. Bull. 53(2010), 385-393

  Published:2010-05-11
 Printed: Sep 2010

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.