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

We formulate a general problem: Given projective schemes $Y$ and $X$ over a global field K and a K-morphism η from $Y$ to $X$ of finite degree, how many points in $X\left(K\right)$ of height at most B have a pre-image under η in $Y\left(K\right)$? 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 $K=F_q\left(T\right)$ and $Y$ is a prime degree cyclic cover of $X=P_K^n$. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.