m at h . A G ] 8 A pr 1 99 8 Gromov - Witten invariants of blow - ups

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 Pr and show that many invariants of these blow-ups can be interpreted as numbers of rational curves on Pr 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.