ag.algebraic geometry - Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?

There is a combinatorial formula for the number of geometric genus $g$ curves of degree $d$ (possibly reducible) passing through $3d-1+g$ generic points of $mathbb{P}^2$ (or a more general toric surface) derived by tropical techniques, see

Grigory Mikhalkin, Enumerative tropical algebraic geometry in $mathbb{R}^2,$
J. Amer. Math. Soc. 18 (2005), no. 2, 313–377 MR