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Science China Mathematics

, Volume 62, Issue 11, pp 2309–2316 | Cite as

Lower degree curves in X3,3 ⊂ ℙ2 × ℙ2

  • Jun LiEmail author
  • Yang Zhou
Articles
  • 17 Downloads

Abstract

In this paper we study low-degree low-genus curves in a generic hypersurface X of degree (3, 3) in ℙ2 × ℙ2. We prove that the genus 0 and genus 1 curves of degree up to (2, 2) are smooth and rigid. We then use the multiple cover formula to compute the Gromov-Witten invariants of X of degree up to (2, 2) and genus up to 2. This provides some initial conditions to determine the full genus 1 and genus 2 Gromov-Witten invariants via Bershadsky-Cecotti-Ooguri-Vafa’s Feynman rule, which is expected to be proved in the near future.

Keywords

Gromov-Witten invariants Calabi-Yau threefolds multiple cover formula 

MSC(2010)

14N35 

Notes

Acknowledgements

The first author was supported by National Science Foundation of USA (Grant Nos. DMS-1564500 and DMS-1601211). The second author was supported by the Simons Collaboration Grant.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Shanghai Center for Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Center of Mathematical Sciences and ApplicationsHarvard UniversityCambridgeUSA

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