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Nodal surfaces with obstructed deformations

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Abstract

In this text we show that the deformation space of a nodal surface X of degree d is smooth and of the expected dimension if \(d\le 7\) or \(d\ge 8\) and X has at most \(4d-5\) nodes (The case \(d\le 7\) was previously covered by Alexandru Dimca using different techniques). For \(d\ge 8\) we give explicit examples of nodal surfaces with \(4d-4\) nodes, for which the tangent space to the deformation space has larger dimension than expected.

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References

  1. Burns Jr., D.M., Wahl, J.M.: Local contributions to global deformations of surfaces. Invent. Math. 26, 67–88 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dimca, A.: Singularities and Topology of Hypersurfaces. Universitext. Springer, New York (1992)

    Book  MATH  Google Scholar 

  3. Dimca, A.: On the syzygies and Hodge theory of nodal hypersurfaces. Preprint arXiv:1310.5344v1 (2013)

  4. Gotzmann, G.: Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes. Math. Z. 158, 61–70 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gourevitch, A., Gourevitch, D.: Geometry of obstructed equisingular families of algebraic hypersurfaces. J. Pure Appl. Algebra 213, 1865–1889 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Green, M.L.: Components of maximal dimension in the Noether-Lefschetz locus. J. Differ. Geom. 29, 295–302 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Greuel, G.-M., Karras, U.: Families of varieties with prescribed singularities. Comp. Math. 69, 83–100 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Kloosterman, R.: Maximal families of nodal varieties with defect. Preprint, arXiv:1310.0227v1 (2013)

  9. Macaulay, F.S.: Some properties of enumeration in the theory of modular systems. Proc. Lond. Math. Soc. S2–26, 531 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhao, Y.: Deformations of nodal surfaces. Ph.D. thesis, Leiden University. https://openaccess.leidenuniv.nl/handle/1887/44549 (2016)

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy

    Remke Kloosterman

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  1. Remke Kloosterman
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Correspondence to Remke Kloosterman.

Additional information

The author would like to thank Arnaud Beauville for pointing out the paper [3]. The author would thank the referee for many valuable suggestions to improve the exposition.

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Kloosterman, R. Nodal surfaces with obstructed deformations. Geom Dedicata 190, 143–150 (2017). https://doi.org/10.1007/s10711-017-0232-2

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  • DOI: https://doi.org/10.1007/s10711-017-0232-2

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