  In this note, we study the fluctuations in the number of points of smooth
projective plane curves over finite fields $\mathbb{F}_q$ as $q$ is fixed and
the genus varies. More precisely, we show that these fluctuations are predicted
by a natural probabilistic model, in which the points of the projective plane
impose independent conditions on the curve. The main tool we use is a geometric
sieving process introduced by Poonen.
