摘要
In this paper we study low-degree low-genus curves in a generic hypersurface X of degree(3, 3) in P^2× P^2. 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.
In this paper we study low-degree low-genus curves in a generic hypersurface X of degree(3, 3) in P^2× P^2. 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.
基金
supported by National Science Foundation of USA (Grant Nos. DMS-1564500 and DMS-1601211)
supported by the Simons Collaboration Grant
