  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.

Algebraic Geometry