Abstract view
Exceptional Covers of Surfaces


Published:20100511
Printed: Sep 2010
Jeffrey D. Achter,
Department of Mathematics, Colorado State University, Fort Collins, CO, USA
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.