Let $k$ be a finite field. We define $N(d,g)$ to be the number of plane curves $f(x,y)$ defined over $k$ of degree $d$ with (geometric) genus $g$. If $D(d) := (d-1)(d-2)/2$ (the maximum possible genus), I would expect that as $d$ goes to infinity that the proportion of curves of degree $d$ with genus $D(d)$ would go to 1. If, on the other hand, we're interested in curves of a fixed small genus (say $g=0$ or 1), I would expect that $N(d,g)$ would still approach infinity, albeit at a much slower rate. The question that I have is how do $N(d,0)$ or $N(d,1)$ approach infinity? Is it a polynomial in $\log d$, faster, slower?
I realize that there has been a lot done with enumerative geometry (Gromon-Witten, Caporaso-Harris), but that seems a bit different, since it always works over an algebraically closed field, and classifies curves by having them pass through some set of generic points, and possibly prescribing, tangency, etc.