Gromov-Witten invariants of blow-ups

TL;DR

This paper provides an explicit algorithm for computing genus zero Gromov-Witten invariants of blow-ups of convex projective varieties, with applications to rational curves and classical multisecant formulas.

Contribution

It introduces a new algorithm for calculating Gromov-Witten invariants of blow-ups and interprets many invariants as counts of rational curves with specific properties.

Findings

Algorithm for computing invariants of blow-ups of convex varieties

Numerical examples including multiplicity d^{-3} for rational curves on the quintic

Connections to classical multisecant formulas

Abstract

In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the original variety. In the second part, we specialize to blow-ups of P^r and show that many invariants of these blow-ups can be interpreted as numbers of rational curves on P^r having specified global multiplicities or tangent directions in the blown-up points. We give various numerical examples, including a new easy way to determine the famous multiplicity d^{-3} for d-fold coverings of rational curves on the quintic threefold, and, as an outlook, two examples of blow-ups along subvarieties, whose Gromov-Witten invariants lead to classical multisecant formulas.