  Let $F$ be a finite field with characteristic greater than two. Define a
\emph{Besicovitch set} in $F^4$ to be a set $P \subseteq F^4$ containing a line
in every direction. The \emph{Kakeya conjecture} asserts that $|P| \approx
|F|^4$. A result of Wolff establishes that $|P| \gtrsim |F|^3$. In this paper
we improve this to $|P| \gtrapprox |F|^{3+\gain}$. On the other hand, we show
that the bound of $|F|^3$ is sharp if we relax the assumption that the lines
point in different directions. One new feature in the argument is the
introduction of a small amount of basic algebraic geometry.
