Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds

Details Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds

1999-11-09

arxiv.org/abs/math/9911056v1

Mathematics

Algebraic Geometry

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Scientific paper

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, and 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.

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