# Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?

I wanted to know if there is something analogous to Kontsevich's recursion formula for enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves. There is a similar formula for genus one curves. See the book "Mirror Symmetry and Algebraic Geometry" by Katz, Page 211.

Any partial results known for g>1? That is, maybe its not known for all d, but for some small values of d?

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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