Gromov-Witten invariants of blow-ups

In the first part of the paper, we give an explicit algorithm to compute th e (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary smooth 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 t he number of rational curves on P-r having specified global multiplicities in the blown-up points.