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 (0l).
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.