Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:36:49Z http://mathoverflow.net/feeds/question/35553 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35553/is-anything-known-about-the-enumeration-of-degree-d-genus-g-curves-in-cp2-where Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ? Ritwik 2010-08-14T02:22:31Z 2010-08-14T09:40:45Z <p>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.</p> <p>Any partial results known for g>1? That is, maybe its not known for all d, but for some small values of d? </p> http://mathoverflow.net/questions/35553/is-anything-known-about-the-enumeration-of-degree-d-genus-g-curves-in-cp2-where/35564#35564 Answer by Victor Protsak for Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ? Victor Protsak 2010-08-14T07:12:59Z 2010-08-14T07:12:59Z <p>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</p> <p>Grigory Mikhalkin, <em>Enumerative tropical algebraic geometry in \$\mathbb{R}^2,\$</em> J. Amer. Math. Soc. 18 (2005), no. 2, 313&ndash;377 <a href="http://www.ams.org/mathscinet-getitem?mr=2137980" rel="nofollow">MR</a></p> http://mathoverflow.net/questions/35553/is-anything-known-about-the-enumeration-of-degree-d-genus-g-curves-in-cp2-where/35570#35570 Answer by Angelo for Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ? Angelo 2010-08-14T09:40:45Z 2010-08-14T09:40:45Z <p>The formula is due to Caporaso and Harris, Counting plane curves of any genus, Invent. Math. 131 (1998), no. 2, 345-392, <a href="http://arxiv.org/abs/alg-geom/9608025" rel="nofollow">http://arxiv.org/abs/alg-geom/9608025</a></p>