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.
- X. Benveniste, Sur le cone des $1$-cycles effectifs en dimension $3$, Math. Ann. 272 (1985), no. 2, 257–265 (French). MR 796252, DOI https://doi.org/10.1007/BF01450570
- Ivan Cheltsov, Nonrational nodal quartic threefolds, Pacific J. Math. 226 (2006), no. 1, 65–81. MR 2247856, DOI https://doi.org/10.2140/pjm.2006.226.65
- Cinzia Casagrande, Priska Jahnke, and Ivo Radloff, On the Picard number of almost Fano threefolds with pseudo-index $>1$, Internat. J. Math. 19 (2008), no. 2, 173–191. MR 2384898, DOI https://doi.org/10.1142/S0129167X08004625
- C. Herbert Clemens, Double solids, Adv. in Math. 47 (1983), no. 2, 107–230. MR 690465, DOI https://doi.org/10.1016/0001-8708%2883%2990025-7
- Alessio Corti, Del Pezzo surfaces over Dedekind schemes, Ann. of Math. (2) 144 (1996), no. 3, 641–683. MR 1426888, DOI https://doi.org/10.2307/2118567
- Steven Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), no. 3, 521–525. MR 936328, DOI https://doi.org/10.1007/BF01456342
- Sławomir Cynk, Defect of a nodal hypersurface, Manuscripta Math. 104 (2001), no. 3, 325–331. MR 1828878, DOI https://doi.org/10.1007/s002290170030
- Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
- Alexandru Dimca, Betti numbers of hypersurfaces and defects of linear systems, Duke Math. J. 60 (1990), no. 1, 285–298. MR 1047124, DOI https://doi.org/10.1215/S0012-7094-90-06010-7
- A. J. de Jong, N. I. Shepherd-Barron, and A. Van de Ven, On the Burkhardt quartic, Math. Ann. 286 (1990), no. 1-3, 309–328. MR 1032936, DOI https://doi.org/10.1007/BF01453578
- Stephan Endraß, On the divisor class group of double solids, Manuscripta Math. 99 (1999), no. 3, 341–358. MR 1702593, DOI https://doi.org/10.1007/s002290050177
- Robert Friedman, Simultaneous resolution of threefold double points, Math. Ann. 274 (1986), no. 4, 671–689. MR 848512, DOI https://doi.org/10.1007/BF01458602
- Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR 0260733, DOI https://doi.org/10.2307/1970746
- J. William Hoffman and Steven H. Weintraub, The Siegel modular variety of degree two and level three, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3267–3305. MR 1828606, DOI https://doi.org/10.1090/S0002-9947-00-02675-1
- V. A. Iskovskih, Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 3, 516–562, 717 (Russian). MR 463151
- V. A. Iskovskih, Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 3, 506–549 (Russian). MR 503430
- Anne-Sophie Kaloghiros. A classification of terminal quartic $3$-folds and applications to rationality questions. Preprint, arxiv:0908.0289.
- Anne-Sophie Kaloghiros. The topology of terminal quartic $3$-folds. University of Cambridge PhD Thesis, arXiv:0707.1852.
- Yujiro Kawamata, Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2) 127 (1988), no. 1, 93–163. MR 924674, DOI https://doi.org/10.2307/1971417
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI https://doi.org/10.2969/aspm/01010283
- Joseph M. Landsberg and Laurent Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. 78 (2003), no. 1, 65–100. MR 1966752, DOI https://doi.org/10.1007/s000140300003
- Massimiliano Mella, Birational geometry of quartic 3-folds. II. The importance of being $\Bbb Q$-factorial, Math. Ann. 330 (2004), no. 1, 107–126. MR 2091681, DOI https://doi.org/10.1007/s00208-004-0542-1
- Shigefumi Mori and Shigeru Mukai, Classification of Fano $3$-folds with $B_2\geq 2$. I, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 496–545. MR 1102273
- Shigefumi Mori and Shigeru Mukai, Classification of Fano $3$-folds with $B_{2}\geq 2$, Manuscripta Math. 36 (1981/82), no. 2, 147–162. MR 641971, DOI https://doi.org/10.1007/BF01170131
- Shigeru Mukai, New developments in Fano manifold theory related to the vector bundle method and moduli problems, Sūgaku 47 (1995), no. 2, 125–144 (Japanese). MR 1364825
- Yoshinori Namikawa, Smoothing Fano $3$-folds, J. Algebraic Geom. 6 (1997), no. 2, 307–324. MR 1489117
- Yoshinori Namikawa and J. H. M. Steenbrink, Global smoothing of Calabi-Yau threefolds, Invent. Math. 122 (1995), no. 2, 403–419. MR 1358982, DOI https://doi.org/10.1007/BF01231450
- Yu. G. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Mat. Sb. 196 (2005), no. 1, 81–122 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 1-2, 77–114. MR 2141325, DOI https://doi.org/10.1070/SM2005v196n01ABEH000873
- Miles Reid. Projective morphisms according to kawamata. Warwick online preprint, 1983.
- B. Saint-Donat, Projective models of $K-3$ surfaces, Amer. J. Math. 96 (1974), 602–639. MR 364263, DOI https://doi.org/10.2307/2373709
- Kil-Ho Shin, $3$-dimensional Fano varieties with canonical singularities, Tokyo J. Math. 12 (1989), no. 2, 375–385. MR 1030501, DOI https://doi.org/10.3836/tjm/1270133187
- Hiromichi Takagi, Classification of primary $\Bbb Q$-Fano threefolds with anti-canonical Du Val $K3$ surfaces. I, J. Algebraic Geom. 15 (2006), no. 1, 31–85. MR 2177195, DOI https://doi.org/10.1090/S1056-3911-05-00416-9
- 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 (1983), no. 6, 1294–1297 (Russian). MR 712934
- Jonathan Wahl, Nodes on sextic hypersurfaces in ${\bf P}^3$, J. Differential Geom. 48 (1998), no. 3, 439–444. MR 1638049
Additional Information
Anne-Sophie Kaloghiros
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
MR Author ID:
912655
ORCID:
0000-0002-8305-8229
Email:
A.S.Kaloghiros@dpmms.cam.ac.uk
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