Skip to main content
SpringerLink
Log in

On the Locus of Nonrigid Hypersurfaces

  • Conference paper
  • First Online:
Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

We show that the Zariski closure of the set of hypersurfaces of degree M in \(\mathbb{P}^{M}\), where M ≥ 5, which are either not factorial or not birationally superrigid, is of codimension at least \(\binom{M - 3}{2} + 1\) in the parameter space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. F. Call, G. Lyubeznik, A simple proof of Grothendieck’s theorem on the parafactoriality of local rings. Contemp. Math. 159, 15–18 (1994)

    Article  MathSciNet  Google Scholar 

  2. I.A. Cheltsov, A double space with a double line. Sb. Math. 195(9–10), 1503–1544 (2004)

    Article  MathSciNet  Google Scholar 

  3. I.A. Cheltsov, On nodal sextic fivefold. Math. Nachr. 280(12), 1344–1353 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Corti, Singularities of linear systems and 3-fold birational geometry, in Explicit Birational Geometry of 3-Folds (Cambridge University Press, Cambridge, 2000), pp. 259–312

    Google Scholar 

  5. A. Corti, M. Mella, Birational geometry of terminal quartic 3-folds. I. Am. J. Math. 126(4), 739–761 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. T. De Fernex, Birational geometry of singular Fano hypersurfaces (2012, preprint) [arXiv:1208.6073]

    Google Scholar 

  7. A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). Documents Mathématiques (Paris) 4, Société Mathématique de France (2005)

    Google Scholar 

  8. J. Harris, in Algebraic Geometry. Graduate Texts in Mathematics, vol. 133 (Springer, New York, 1992)

    Google Scholar 

  9. R. Hartshorne, in Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977)

    Google Scholar 

  10. V.A. Iskovskikh, Yu.I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem. Math. USSR Sb. 86(1), 140–166 (1971)

    Google Scholar 

  11. M. Mella, Birational geometry of quartic 3-folds. II. Math. Ann. 330(1), 107–126 (2004)

    MATH  MathSciNet  Google Scholar 

  12. R. Mullany, Fano double spaces with a big singular locus. Math. Notes 87(3), 444–448 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. A.V. Pukhlikov, Birational automorphisms of a three-dimensional quartic with an elementary singularity. Math. USSR Sb. 63, 457–482 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. A.V. Pukhlikov, Birational automorphisms of Fano hypersurfaces. Invent. Math. 134(2), 401–426 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. A.V. Pukhlikov, Essentials of the method of maximal singularities, in Explicit Birational Geometry of 3-Folds (Cambridge University Press, Cambridge, 2000), pp. 73–100

    Google Scholar 

  16. A.V. Pukhlikov, Birationally rigid Fano complete intersections. Crelle J. für die reine und angew. Math. 541, 55–79 (2001)

    MATH  MathSciNet  Google Scholar 

  17. A.V. Pukhlikov, Birationally rigid Fano hypersurfaces with isolated singularities. Sb. Math. 193(3), 445–471 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. A.V. Pukhlikov, Birationally rigid singular Fano hypersurfaces. J. Math. Sci. 115(3), 2428–2436 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. A.V. Pukhlikov, Birationally rigid varieties, I. Fano varieties. Russ. Math. Surv. 62(5), 857–942 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. A.V. Pukhlikov, Birational geometry of singular Fano varieties. Proc. Steklov Inst. Math. 264, 159–177 (2009)

    Article  MathSciNet  Google Scholar 

  21. A.V. Pukhlikov, Birationally Rigid Varieties. Mathematical Surveys and Monographs, vol. 190 (American Mathematical Society, Providence, 2013)

    Google Scholar 

  22. K. Shramov, Birational automorphisms of nodal quartic threefolds (2008) [arXiv:0803.4348]

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, M &O Building, Peach Street, Liverpool, L69 7ZL, UK

    Thomas Eckl & Aleksandr Pukhlikov

Authors
  1. Thomas Eckl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aleksandr Pukhlikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aleksandr Pukhlikov .

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

    Ivan Cheltsov

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Ciro Ciliberto

  3. Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

    Hubert Flenner

  4. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    James McKernan

  5. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri G. Prokhorov

  6. Institute Fourier, UMR 5582, CNRS-UJF, University of Grenoble I, Saint Martin D’Hères, France

    Mikhail Zaidenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Eckl, T., Pukhlikov, A. (2014). On the Locus of Nonrigid Hypersurfaces. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation