  We prove a new upper bound for the number of incidences between points and
lines in a plane over an arbitrary field $\mathbb{F}$, a problem first
considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and
$n$ lines in $\mathbb{F}^2$, with $m^{7/8}<n<m^{8/7}$, determine at most
$O(m^{11/15}n^{11/15})$ incidences (where, if $\mathbb{F}$ has positive
characteristic $p$, we assume $m^{-2}n^{13}\ll p^{15}$). This improves on the
previous best known bound, due to Jones. To obtain our bound, we first prove an
optimal point-line incidence bound on Cartesian products, using a reduction to
a point-plane incidence bound of Rudnev. We then cover most of the point set
with Cartesian products, and we bound the incidences on each product
separately, using the bound just mentioned. We give several applications, to
sum-product-type problems, an expander problem of Bourgain, the distinct
distance problem and Beck's theorem.
