Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The defect of Fano $ 3$-folds


Author: Anne-Sophie Kaloghiros
Journal: J. Algebraic Geom. 20 (2011), 127-149
DOI: https://doi.org/10.1090/S1056-3911-09-00531-1
Published electronically: October 7, 2009
Erratum: J. Algebraic Geom. 21 (2012) 397-399.
MathSciNet review: 2729277
Full-text PDF

Abstract | References | Additional Information

Abstract: This paper studies the rank of the divisor class group of terminal Gorenstein Fano $ 3$-folds. If $ Y$ is not $ \mathbb{Q}$-factorial, there is a small modification of $ Y$ with a second extremal ray; Cutkosky, following Mori, gave an explicit geometric description of contractions of extremal rays on terminal Gorenstein $ 3$-folds. I introduce the category of weak-star Fanos, which allows one to run the Minimal Model Program (MMP) in the category of Gorenstein weak Fano $ 3$-folds. If $ Y$ does not contain a plane, the rank of its divisor class group can be bounded by running an MMP on a weak-star Fano small modification of $ Y$. These methods yield more precise bounds on the rank of $ \operatorname{Cl} Y$ depending on the Weil divisors lying on $ Y$. I then study in detail quartic $ 3$-folds that contain a plane and give a general bound on the rank of the divisor class group of quartic $ 3$-folds. Finally, I indicate how to bound the rank of the divisor class group of higher genus terminal Gorenstein Fano $ 3$-folds with Picard rank $ 1$ that contain a plane.


References [Enhancements On Off] (What's this?)

  • [Ben85] X. Benveniste.
    Sur le cone des $ 1$-cycles effectifs en dimension $ 3$.
    Math. Ann., 272(2):257-265, 1985. MR 796252 (86j:14005)
  • [Che06] Ivan Cheltsov.
    Nonrational nodal quartic threefolds.
    Pacific J. Math., 226(1):65-81, 2006. MR 2247856 (2007e:14024)
  • [CJR08] Cinzia Casagrande, Priska Jahnke, and Ivo Radloff.
    On the Picard number of almost Fano threefolds with pseudo-index $ >1$.
    Internat. J. Math., 19(2):173-191, 2008. MR 2384898 (2008m:14080)
  • [Cle83] C. Herbert Clemens.
    Double solids.
    Adv. in Math., 47(2):107-230, 1983. MR 690465 (85e:14058)
  • [Cor96] Alessio Corti.
    Del Pezzo surfaces over Dedekind schemes.
    Ann. of Math. (2), 144(3):641-683, 1996. MR 1426888 (98e:14037)
  • [Cut88] Steven Cutkosky.
    Elementary contractions of Gorenstein threefolds.
    Math. Ann., 280(3):521-525, 1988. MR 936328 (89k:14070)
  • [Cyn01] Sławomir Cynk.
    Defect of a nodal hypersurface.
    Manuscripta Math., 104(3):325-331, 2001. MR 1828878 (2002g:14056)
  • [Del74] Pierre Deligne.
    Théorie de Hodge. III.
    Inst. Hautes Études Sci. Publ. Math., (44):5-77, 1974. MR 0498552 (58:16653b)
  • [Dim90] Alexandru Dimca.
    Betti numbers of hypersurfaces and defects of linear systems.
    Duke Math. J., 60(1):285-298, 1990. MR 1047124 (91f:14041)
  • [dJSBVdV90] A. J. de Jong, N. I. Shepherd-Barron, and A. Van de Ven.
    On the Burkhardt quartic.
    Math. Ann., 286(1-3):309-328, 1990. MR 1032936 (91f:14038)
  • [End99] Stephan Endraß.
    On the divisor class group of double solids.
    Manuscripta Math., 99(3):341-358, 1999. MR 1702593 (2000f:14058)
  • [Fri86] Robert Friedman.
    Simultaneous resolution of threefold double points.
    Math. Ann., 274(4):671-689, 1986. MR 848512 (87k:32035)
  • [Gri69] Phillip A. Griffiths.
    On the periods of certain rational integrals. I, II.
    Ann. of Math. (2) 90 (1969), 460-495; ibid. (2), 90:496-541, 1969. MR 0260733 (41:5357)
  • [HW01] J. William Hoffman and Steven H. Weintraub.
    The Siegel modular variety of degree two and level three.
    Trans. Amer. Math. Soc., 353(8):3267-3305 (electronic), 2001. MR 1828606 (2003b:11044)
  • [Isk77] V. A. Iskovskih.
    Fano threefolds. I.
    Izv. Akad. Nauk SSSR Ser. Mat., 41(3):516-562, 717, 1977. MR 463151 (80c:14023a)
  • [Isk78] V. A. Iskovskih.
    Fano threefolds. II.
    Izv. Akad. Nauk SSSR Ser. Mat., 42(3):506-549, 1978. MR 503430 (80c:14023b)
  • [Kal] Anne-Sophie Kaloghiros.
    A classification of terminal quartic $ 3$-folds and applications to rationality questions.
    Preprint, arxiv:0908.0289.
  • [Kal07] Anne-Sophie Kaloghiros.
    The topology of terminal quartic $ 3$-folds.
    University of Cambridge PhD Thesis, arXiv:0707.1852.
  • [Kaw88] Yujiro Kawamata.
    Crepant blowing-up of $ 3$-dimensional canonical singularities and its application to degenerations of surfaces.
    Ann. of Math. (2), 127(1):93-163, 1988. MR 924674 (89d:14023)
  • [KMM87] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki.
    Introduction to the minimal model problem.
    In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987. MR 946243 (89e:14015)
  • [LM03] Joseph M. Landsberg and Laurent Manivel.
    On the projective geometry of rational homogeneous varieties.
    Comment. Math. Helv., 78(1):65-100, 2003. MR 1966752 (2004a:14050)
  • [Mel04] Massimiliano Mella.
    Birational geometry of quartic 3-folds. II. The importance of being $ \mathbb{Q}$-factorial.
    Math. Ann., 330(1):107-126, 2004. MR 2091681 (2005h:14030)
  • [MM86] Shigefumi Mori and Shigeru Mukai.
    Classification of Fano $ 3$-folds with $ B\sb 2\geq 2$. I.
    In Algebraic and topological theories (Kinosaki, 1984), pages 496-545. Kinokuniya, Tokyo, 1986. MR 1102273
  • [MM82] Shigefumi Mori and Shigeru Mukai.
    Classification of Fano $ 3$-folds with $ B\sb{2}\geq 2$.
    Manuscripta Math., 36(2):147-162, 1981/82. MR 641971 (83f:14032)
  • [Muk02] Shigeru Mukai.
    New developments in the theory of Fano threefolds: vector bundle method and moduli problems [translation of Sūgaku 47 (1995), no. 2, 125-144].
    Sugaku Expositions, 15(2):125-150, 2002.
    Sugaku expositions. MR 1364825 (96m:14059)
  • [Nam97] Yoshinori Namikawa.
    Smoothing Fano $ 3$-folds.
    J. Algebraic Geom., 6(2):307-324, 1997. MR 1489117 (99d:14040)
  • [NS95] Yoshinori Namikawa and J. H. M. Steenbrink.
    Global smoothing of Calabi-Yau threefolds.
    Invent. Math., 122(2):403-419, 1995. MR 1358982 (96m:14056)
  • [Pro05] Yu. G. Prokhorov.
    The degree of Fano threefolds with canonical Gorenstein singularities.
    Mat. Sb., 196(1):81-122, 2005. MR 2141325 (2006e:14058)
  • [Rei83] Miles Reid.
    Projective morphisms according to kawamata.
    Warwick online preprint, 1983.
  • [SD74] B. Saint-Donat.
    Projective models of K$ 3$ surfaces.
    Amer. J. Math., 96:602-639, 1974. MR 0364263 (51:518)
  • [Shi89] Kil-Ho Shin.
    $ 3$-dimensional Fano varieties with canonical singularities.
    Tokyo J. Math., 12(2):375-385, 1989. MR 1030501 (90j:14050)
  • [Tak06] Hiromichi Takagi.
    Classification of primary $ \mathbb{Q}$-Fano threefolds with anti-canonical Du Val $ K3$ surfaces. I.
    J. Algebraic Geom., 15(1):31-85, 2006. MR 2177195 (2006k:14071)
  • [Var83] A. N. Varchenko.
    Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface.
    Dokl. Akad. Nauk SSSR, 270(6):1294-1297, 1983. MR 712934 (85d:32028)
  • [Wah98] Jonathan Wahl.
    Nodes on sextic hypersurfaces in $ \mathbb{P}^ 3$.
    J. Differential Geom., 48(3):439-444, 1998. MR 1638049 (99g:14055)


Additional Information

Anne-Sophie Kaloghiros
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: A.S.Kaloghiros@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S1056-3911-09-00531-1
Received by editor(s): August 5, 2008
Received by editor(s) in revised form: February 24, 2009
Published electronically: October 7, 2009
Additional Notes: This work was partially supported by Trinity Hall, Cambridge

American Mathematical Society