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Exceptional Covers of Surfaces
Published online by Cambridge University Press: 20 November 2018
Abstract
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.
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- Copyright © Canadian Mathematical Society 2010