Counting curves of any genus on P²₆

2011/06/30 by Shoval, M., Shustin, E.
#14H20 #14J26 (Secondary) #14N35 (Primary) 14H10 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1106.6155

Abstract

We obtain a formula for the degrees of the varieties parameterizing complex algebraic curves of any divisor class and genus on P²₆, the projective plane blown-up at 6 generic points. Moreover, the formula computes the degrees of the varieties parameterizing curves on P²₆ which additionally satisfy certain tangency conditions to a fixed exceptional divisor on P²₆. Our formula contains as special cases the degrees of the analogous varieties parameterizing curves on P²_q, for q=1,...,5, and on the quadric (P¹)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 P²₆.

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