Tropical enumeration of curves in blowups of the projective plane

Abstract

We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recursive computation of Gromov-Witten invariants relative a normal crossing divisor.

25 pages, 22 pictures. A talk with many more pictures, and a Mathematica program computing these invariants is available on my website: https://sites.google.com/view/brettparker/publications. v2: minor improvements in exposition, and updated references. v3: An expanded, more formal version from 2019, more accurately reflecting the published version