## 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 Σ_{n|d}1/n^{3}. (2) For a smoothly embedded contractable curve C ⊂ Y we define schemes C_{i} for 1 ≤ i ≤ l where C_{i} is supported on C and has multiplicity i, the number l ∈ {1,...,6} being Kollár's invariant "length". We prove that the contribution of multiple covers of C of degree d is given by Σ_{n|d}kd/n/n^{3}, 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, which are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers.

Original language | English (US) |
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Pages (from-to) | 549-568 |

Number of pages | 20 |

Journal | Journal of Algebraic Geometry |

Volume | 10 |

Issue number | 3 |

State | Published - Jul 1 2001 |

Externally published | Yes |

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology