Geometric generalizations of the square sieve, with an application to cyclic covers

We formulate a general problem: Given projective schemes Y$\mathbb {Y}$ and X$\mathbb {X}$ over a global field K and a K-morphism eta from Y$\mathbb {Y}$ to X$\mathbb {X}$ of finite degree, how many points in X(K)$\mathbb {X}(K)$ of height at most B have a pre-image under eta in Y(K)$\mathbb {Y}(K)$? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when K=Fq(T)$K=\mathbb {F}_q(T)$ and Y$\mathbb {Y}$ is a prime degree cyclic cover of X=PKn$\mathbb {X}=\mathbb {P}_{K}<^>n$. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.