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Singular cubics of CP 2 as limits of sequences of tori

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Abstract

We use the modular invariant j to understand singular cubics of CP 2 as limits of sequences of tori and we observe different behaviours according to the cubic type.

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References

  1. Brieskorn E. and Knörrer H. (1986). Plane Algebraic Curves. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  2. Gromov M. (1985). Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82: 307–347

    Article  MATH  MathSciNet  Google Scholar 

  3. Hilbert D. (1890). Ueber die Theorie der algebraischen Formen. Math. Ann. 36: 473–534

    Article  MathSciNet  Google Scholar 

  4. Hilbert D. (1893). Ueber die vollen Invariantensysteme. Math. Ann. 42: 313–373

    Article  MathSciNet  Google Scholar 

  5. Hilbert D. (1993). Theory of Algebraic Invariants. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  6. Kodaira K. (1963). On compact analytic surfaces II. Ann. of Math. 77: 563–626

    Article  Google Scholar 

  7. Kra I. (1990). Teichmüller and Riemann spaces of Kleinian groups. J. Am. Math. Soc. 3: 499–578

    Article  MATH  MathSciNet  Google Scholar 

  8. Kraft H. (1985). Geometrische Methoden in Der Invariantentheorie. Friedr. Vieweg & Sohn, Braunschweig Wiesbaden

    MATH  Google Scholar 

  9. McDuff, D., Salamon, D.: J-holomorphic Curves and Symplectic Topology. American Mathematical Society. Colloquium publications 52, Providence, Rhode Island (2004)

  10. Muller M.-P. (1994). Gromov’s Schwarz lemma as an estimate of the gradient for holomorphic curves. In: Audin, M. and Lafontaine, J. (eds) Holomorphic Curves in Symplectic. Geometry Progress in Mathematics, vol. 117., pp 217–231. Birkhäuser Verlag, Basel

    Google Scholar 

  11. Mumford, D.: Stability of projective varieties. Enseignement Math. 24 (1977)

  12. Mumford D. and Fogarty J. (1982). Geometric Invariant Theory. Second enlarged edition. Springer-Verlag, Berlin

    Google Scholar 

  13. Nakamura I. (2004). Planar cubic curves—from Hesse to Mumford, Sagaku Expositions 17(1): 73–101

    Google Scholar 

  14. Parker T. and Wolfson J. (1993). Pseudo-holomorphic maps and bubble trees. J. Geom. Anal. 3: 63–98

    MATH  MathSciNet  Google Scholar 

  15. Serre, J.-P.: Cours d’arithmétique. Presses universitaires de France. Le mathématicien (1988)

  16. Witten E. (1994). Monopoles and four-manifolds. Math. Res. Lett. 1: 769–796

    MATH  MathSciNet  Google Scholar 

  17. Ye R. (1994). Gromov’s compactness theorem for pseudo-holomorphic curves. Trans. Am. Math. Soc. 342: 671–694

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. I.U.F.M. Midi-Pyrénées/Site de Rangueil, 118, route de Narbonne, 31078, Toulouse cedex 04, France

    V. Lizan-Esquerrétou

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  1. V. Lizan-Esquerrétou
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Correspondence to V. Lizan-Esquerrétou.

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Lizan-Esquerrétou, V. Singular cubics of CP 2 as limits of sequences of tori. Geom Dedicata 132, 81–93 (2008). https://doi.org/10.1007/s10711-007-9201-5

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  • DOI: https://doi.org/10.1007/s10711-007-9201-5

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