Summary: EXCEPTIONAL COVERS OF SURFACES
JEFFREY D. ACHTER
ABSTRACT. Consider a finite morphism f : X Y of smooth, projective varieties over a finite field
F. Suppose X is the vanishing locus in PN 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 |F| > C, then f (F) : X(F) Y(F) is
injective if and only if it is surjective.
Consider a finite, generically ´etale morphism f : X Y between smooth, projective varieties
over a finite field F of characteristic p. The cover f is called exceptional if the only geometrically
irreducible component of X ×Y X which is defined over F is the diagonal. Exceptional covers have
the following intriguing property: the induced map f (F) : X(F) Y(F) on F-points is bijective.
This theorem, due to Lenstra, is proved in ; we defer to that article for the history of this circle
In , Guralnick, Tucker and Zieve prove a partial converse for projective curves. Specifically,
they show that for fixed genus g = g(X) and degree deg(f ), there exists an effective constant C
such that the following holds: if Fq/F is an extension with q > C, and if f (Fq) is injective, then f
is exceptional. (Note that this implies that f is bijective.) They prove something like this in higher
dimension (see Remark 2.6 below), except that the constant C is allowed to depend on X, Y and f.
They conjecture [6, 5.5] that C need only depend on deg(f ) and the topology of X.
The calculation of C relies on understanding the topology of the cover f. Indeed, if Z is a