The average rank of elliptic $n$-folds

Let $V/\mathbb{F}_q$ be a variety of dimension at least two. We show that the density of elliptic curves $E/\mathbb{F}_q(V)$ with positive rank is zero if $V$ has dimension at least 3 and is at most $1-\zeta_V(3)^{-1}$ if $V$ is a surface.