Proceedings of the Steklov Institute of Mathematics

, Volume 299, Issue 1, pp 205–218 | Cite as

Factorial Hypersurfaces

Article

Abstract

The codimension of the complement of the set of factorial hypersurfaces of degree d in PN is estimated for d ≥ 4 and N ≥ 7.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. Call and G. Lyubeznik, “A simple proof of Grothendieck’s theorem on the parafactoriality of local rings,” in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (Am. Math. Soc., Providence, RI, 1994), Contemp. Math. 159, pp. 15–18.Google Scholar
  2. 2.
    I. A. Cheltsov, “Factoriality of nodal three-dimensional varieties and connectedness of the locus of log canonical singularities,” Mat. Sb. 197 (3), 87–116 (2006) [Sb. Math. 197, 387–414 (2006)].MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    I. Cheltsov, “Factorial threefold hypersurfaces,” J. Algebr. Geom. 19 (4), 781–791 (2010).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    I. Cheltsov and J. Park, “Factorial hypersurfaces in P4 with nodes,” Geom. Dedicata 121, 205–219 (2006).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    C. Ciliberto and V. Di Gennaro, “Factoriality of certain threefolds complete intersection in P5 with ordinary double points,” Commun. Algebra 32 (7), 2705–2710 (2004).CrossRefMATHGoogle Scholar
  6. 6.
    M. Mella, “Birational geometry of quartic 3-folds. II: The importance of being Q-factorial,” Math. Ann. 330 (1), 107–126 (2004).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    F. Polizzi, A. Rapagnetta, and P. Sabatino, “On factoriality of threefolds with isolated singularities,” Mich. Math. J. 63 (4), 781–801 (2014).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    A. V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces,” Invent. Math. 134 (2), 401–426 (1998).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A. V. Pukhlikov, “Birationally rigid Fano complete intersections,” J. Reine Angew. Math. 541, 55–79 (2001).MathSciNetMATHGoogle Scholar
  10. 10.
    A. Pukhlikov, Birationally Rigid Varieties (Am. Math. Soc., Providence, RI, 2013), Math. Surv. Monogr. 190.CrossRefMATHGoogle Scholar
  11. 11.
    A. V. Pukhlikov, “Birational geometry of Fano hypersurfaces of index two,” Math. Ann. 366 (1–2), 721–782 (2016).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    A. V. Pukhlikov, “Birational geometry of algebraic varieties fibred into Fano double spaces,” Izv. Ross. Akad. Nauk, Ser. Mat. 81 (3), 160–188 (2017) [Izv. Math. 81, 618–644 (2017)].MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesThe University of Liverpool, Mathematical Sciences BuildingLiverpoolUK

Personalised recommendations