  We study Gromov-Witten invariants on the blow-up of P^n at a point, which is
probably the simplest example of a variety whose moduli spaces of stable maps
do not have the expected dimension. It is shown that many of these invariants
can be interpreted geometrically on P^n as certain numbers of rational curves
having a multiple point of given order at the blown up point. Moreover, all
these invariants can actually be calculated, giving enumerative invariants of
P^n which have not been known before.
