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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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The formula is due to Caporaso and Harris, Counting plane curves of any genus, Invent. Math. 131 (1998), no. 2, 345-392,

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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

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