  We propose an algebraic geometry framework for the Kakeya problem. We
conjecture that for any polynomials $f,g\in\F_{q_0}[x,y]$ and any
$\F_q/\F_{q_0}$, the image of the map $\F_q^3\to\F_q^3$ given by
$(s,x,y)\mapsto (s,sx+f(x,y),sy+g(x,y))$ has size at least
$\frac{q^3}{4}-O(q^{5/2})$ and prove the special case when $f=f(x), g=g(y).$ We
also prove it in the case $f=f(y), g=g(x)$ under the additional assumption
$f'(0)g'(0)\neq 0$ when $f,g$ are both linearized. Our approach is based on a
combination of Cauchy--Schwarz and Lang--Weil. The algebraic geometry inputs in
the proof are various results concerning irreducibility of certain classes of
multivariate polynomials.
