Lower degree curves in X3,3 ⊂ ℙ2 × ℙ2
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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.
KeywordsGromov-Witten invariants Calabi-Yau threefolds multiple cover formula
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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