  We formulate a general problem: given projective schemes $\mathbb{Y}$ and
$\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from
$\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in
$\mathbb{X}(K)$ of height at most $B$ have a pre-image under $\eta$ in
$\mathbb{Y}(K)$? 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
non-trivial answer to the general problem when $K=\mathbb{F}_q(T)$ and
$\mathbb{Y}$ is a prime degree cyclic cover of $\mathbb{X}=\mathbb{P}_{K}^n$.
Our tool is a new geometric sieve, which generalizes the polynomial sieve to a
geometric setting over global function fields.
