Counting curves of any genus on P^2_6

Details Counting curves of any genus on P^2_6 Counting curves of any genus on P^2_6

2011-06-30

arxiv.org/abs/1106.6155v4

Mathematics

Algebraic Geometry

54 pages, 3 figures, typos corrected

Scientific paper

We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P^2_6, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the varieties parameterizing curves on P^2_6 which additionally satisfy certain tangency conditions to a fixed exceptional divisor on P^2_6. Our formula contains as special cases the degrees of the analogous varieties parameterizing curves on P^2_q, for q=1,...,5, and on the quadric (P^1)^2. It is an extension of the Caporaso-Harris formula counting curves of any degree and genus in the projective plane, and it can be viewed as an alternative to the Vakil formula, which gives an indirect way to count curves of any genus on P^2_6.

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