Abstract: 

We study the contribution of multiple covers of an irreducible rational
curve C in a Calabi-Yau threefold Y to the genus 0 Gromov-Witten
invariants in the following cases.


(1) If the curve C has one node and satisfies a certain genericity
condition, we prove that the contribution of multiple covers of degree d is
given by the sum of all 1/n^3 where n divides d.

(2) For a smoothly embedded contractable curve C in Y we define schemes C_i
for i=1,...,l where C_i is supported on C and has multiplicity i, the
integer l (0<l<7) being Koll\'ar's invariant ``length''.  We prove that the
contribution of multiple covers of C of degree d is given by the sum of
k_{d/n}/n^3 where n divides d and where k_i is the multiplicity of C_i in
its Hilbert scheme (and k_i=0 if i>l).

In the latter case we also get a formula for arbitrary genus.

These results show that the curve C contributes an integer amount to the
so-called instanton numbers that are defined recursively in terms of the
Gromov-Witten invariants and are conjectured to be integers.