Annals of Global Analysis and Geometry

, Volume 39, Issue 3, pp 293–323 | Cite as

Minitwistor spaces, Severi varieties, and Einstein–Weyl structure

Original Paper

Abstract

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein–Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein–Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein–Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.

Keywords

Minitwistor space Twistor space Einstein–Weyl structure Severi variety Nodal rational curve Penrose correspondence 

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References

  1. 1.
    Birth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge Band 4, Springer-Verlag, BerlinGoogle Scholar
  2. 2.
    Calderbank D.M.J., Pedersen H.: Self-dual Einstein metrics with torus symmetry. J. Differential Geom. 60, 485–521 (2002)MATHMathSciNetGoogle Scholar
  3. 3.
    Greuel G., Lossen C., Shustin E.: Geometry of families of nodal curves on the blown-up projective plane. Trans. Amer. Math. Soc. 350, 251–274 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Hitchin N.: Complex manifolds and Einstein’s equations. Lect. Notes Math. 970, 73–99 (1982)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Honda N.: Explicit construction of new Moishezon twistor spaces. J. Differential Geom. 82, 411–444 (2009) math.DG/0701278MATHMathSciNetGoogle Scholar
  6. 6.
    Honda N.: Double solid twistor spaces: the case of arbitrary signature. Invent. Math. 174, 463–504 (2008) arXiv:0705.0060MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Honda N.: A new series of compact minitwistor spaces and Moishezon twistor spaces over them. J. Reine Angew. Math. 642, 197–235 (2010) arXiv:0805.0042MATHMathSciNetGoogle Scholar
  8. 8.
    Jones P.E., Tod K.P.: Minitwistor spaces and Einstein–Weyl geometry. Classical Quantum Gravity 2, 565–577 (1985)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Joyce D.: Explicit construction of self-dual 4-manifolds. Duke Math. J. 77, 519–552 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Nakata F.: A construction of Einstein–Weyl spaces via LeBrun-Mason type twistor correspondence. Comm. Math. Phys. 289, 663–699 (2009) arXiv:0806.2696MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Pedersen H.: Einstein–Weyl spaces and (1, n)-curves in the quadric surface. Ann. Global Anal. Geom. 4, 89–120 (1986)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pedersen H., Tod K.P.: Three-dimensional Einstein–Weyl geometry. Adv. Math. 97, 74–109 (1993)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sernesi E.: Deformations of Algebraic Schemes, Grundlehren der Mathematischen Wissenschaften, vol. 334. Springer-Verlag, Berlin (2006)Google Scholar
  14. 14.
    Wahl J.: Deformations of plane curves with nodes and cusps. Amer. J. Math. 96, 529–577 (1974)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ward R.S.: Einstein–Weyl spaces and SU(∞) Toda fields. Classical Quantum Gravity 7, L95–L98 (1990)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of Science and EngineeringTokyo Institute of TechnologyMeguroJapan
  2. 2.Mathematical Institute, Graduate School of ScienceTohoku UniversitySendaiJapan
  3. 3.Department of Mathematics, Faculty of Science and TechnologyTokyo University of ScienceChibaJapan

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