Counting curves of any genus on P^2_7

Details Counting curves of any genus on P^2_7 Counting curves of any genus on P^2_7

2011-08-15

arxiv.org/abs/1108.3089v3

Mathematics

Algebraic Geometry

54 pages, types corrected, made independent of the preprint at arXiv:1106.6155

Scientific paper

We enumerate complex algebraic curves of any divisor class and genus on P^2_7, the projective plane blown-up at 7 generic points. For a fewer number of blown up points, the result was obtained by Vakil. Our computation is done in two steps: (1) we consider the surfaces P^2_{a,1}, the plane blown up at a>5 points on a conic and one more point outside this conic, and, using techniques of tropical geometry, we obtain a Caporaso-Harris-Vakil type formula counting curves of any divisor class and genus subject to arbitrary tangency conditions with respect to the blown up conic, (2) then we express the Gromov-Witten invariants of P^2_7 via enumerative invariants of P^2_{6,1}, using Vakil's version of Abramovich-Bertram formula.

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