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Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization

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We describe an algebro-geometric approach to Vakil–Zinger’s desingularization of the main component of the moduli of genus one stable maps to \({\mathbb{P}^{n}}\) (Vakil and Zinger in Res Announc Am Math Soc 13:53–59, 2007; Geom Topol 12(1):1–95, 2008). Our approach is based on understanding the local structure of this moduli space; it also gives a partial desingularization of the entire moduli space. The results proved should extend to higher genera.

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References

  1. Chang, H.-L., Li, J.: On reduced genus one GW-invariants of Quintics (2010, in preparation)

  2. Kontsevich M.: Enumeration of rational curves via torus actions. In: Dijkgraaf, R., Faber, C., Geer, G. (eds) The Moduli Space of Curves. Progress in Mathematics, vol. 129, pp. 335–368. Birkhäuser, Basel (1995)

    Google Scholar 

  3. Li J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60(2), 199–293 (2001) MR1938113 (2004k:14096)

    Google Scholar 

  4. Li J., Zinger A.: On the genus-one Gromov–Witten invariants of complete intersections. J. Differ. Geom. 82(3), 641–690 (2009)

    MATH  MathSciNet  Google Scholar 

  5. Pandharipande R.: A note on elliptic plane curves with fixed j-invariant. Proc. Am. Math. Soc. 125(12), 3471–3479 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Vakil R., Zinger A.: A natural smooth compactification of the space of elliptic curves in projective space. Electron. Res. Announc. Am. Math. Soc. 13, 53–59 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Vakil R., Zinger A.: A desingularization of the main component of the moduli space of genus-one stable maps into \({\mathbb{P}^{n}}\). Geom. Topol. 12(1), 1–95 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zinger A.: On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps. Adv. Math. 214(2), 878–933 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zinger A.: The reduced genus-one Gromov–Witten invariants of Calabi–Yau hypersurfaces. J. Am. Math. Soc. 22, 691–737 (2009)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Arizona, USA

    Yi Hu

  2. Department of Mathematics, Stanford University, Stanford, USA

    Jun Li

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  1. Yi Hu
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  2. Jun Li
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Correspondence to Yi Hu.

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Hu, Y., Li, J. Genus-one stable maps, local equations, and Vakil–Zinger’s desingularization. Math. Ann. 348, 929–963 (2010). https://doi.org/10.1007/s00208-010-0504-8

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  • DOI: https://doi.org/10.1007/s00208-010-0504-8

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