Classification of primary $\mathbb {Q}$-Fano threefolds with anti-canonical Du Val $K3$ surfaces, I
Author:
Hiromichi Takagi
Journal:
J. Algebraic Geom. 15 (2006), 31-85
DOI:
https://doi.org/10.1090/S1056-3911-05-00416-9
Published electronically:
June 27, 2005
MathSciNet review:
2177195
Full-text PDF
Abstract | References | Additional Information
Abstract: If a non-Gorenstein $\mathbb {Q}$-Fano threefold with only cyclic quotient terminal singularities has anti-canonical Du Val $K3$ surfaces and the anti-canonical class generates the group of numerical equivalence classes of divisors, then the dimension of the space of global sections of the anti-canonical sheaf is shown to be not greater than ten. Such $\mathbb {Q}$-Fano threefolds with the dimension not less than nine are classified.
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932
- Valery Alexeev, Ample Weil divisors on $K3$ surfaces with Du Val singularities, Duke Math. J. 64 (1991), no. 3, 617–624. MR 1141287, DOI https://doi.org/10.1215/S0012-7094-91-06429-X
- Valery Alexeev, General elephants of ${\bf Q}$-Fano 3-folds, Compositio Math. 91 (1994), no. 1, 91–116. MR 1273928 [Alt]Alt S. Altınok, Graded rings corresponding to polarized ${K}3$ surfaces and $\mathbb {Q}$-fano $3$-folds, Univ. of Warwick, Ph. D. thesis, available at: http://www.maths.warwick.ac.uk/ miles/doctors/Selma/
- M. Andreatta and J. A. Wiśniewski, A note on nonvanishing and applications, Duke Math. J. 72 (1993), no. 3, 739–755. MR 1253623, DOI https://doi.org/10.1215/S0012-7094-93-07228-6
- F. Campana and H. Flenner, Projective threefolds containing a smooth rational surface with ample normal bundle, J. Reine Angew. Math. 440 (1993), 77–98. MR 1225958
- Steven Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Ann. 280 (1988), no. 3, 521–525. MR 936328, DOI https://doi.org/10.1007/BF01456342 [Fan37]Fano G. Fano, Sulle varietà a tre dimensioni a curve-sezioni canoniche, R. Accad. d’Italia Mem. Cl. Sci. Fis. Mat. Nat. 8 (1936/37), 23–64.
- A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR 1798982
- Takao Fujita, On the structure of polarized manifolds with total deficiency one. I, J. Math. Soc. Japan 32 (1980), no. 4, 709–725. MR 589109, DOI https://doi.org/10.2969/jmsj/03240709
- Takao Fujita, On the structure of polarized manifolds with total deficiency one. II, J. Math. Soc. Japan 33 (1981), no. 3, 415–434. MR 620281, DOI https://doi.org/10.2969/jmsj/03330415
- Takao Fujita, On the structure of polarized manifolds with total deficiency one. III, J. Math. Soc. Japan 36 (1984), no. 1, 75–89. MR 723595, DOI https://doi.org/10.2969/jmsj/03610075
- Takao Fujita, Projective varieties of $\Delta $-genus one, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 149–175. MR 1102257 [Fuj90a]Fubook ---, Classification Theories of Polarized Varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, 1990.
- Takao Fujita, On singular del Pezzo varieties, Algebraic geometry (L’Aquila, 1988) Lecture Notes in Math., vol. 1417, Springer, Berlin, 1990, pp. 117–128. MR 1040555, DOI https://doi.org/10.1007/BFb0083337
- N. P. Gushel′, Fano varieties of genus $8$, Uspekhi Mat. Nauk 38 (1983), no. 1(229), 163–164 (Russian). MR 693729
- Takayuki Hayakawa, Blowing ups of $3$-dimensional terminal singularities, Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 515–570. MR 1710753, DOI https://doi.org/10.2977/prims/1195143612
- Takayuki Hayakawa, Blowing ups of 3-dimensional terminal singularities. II, Publ. Res. Inst. Math. Sci. 36 (2000), no. 3, 423–456. MR 1781436, DOI https://doi.org/10.2977/prims/1195142953
- V. A. Iskovskikh, Double projection from a line onto Fano $3$-folds of the first kind, Mat. Sb. 180 (1989), no. 2, 260–278 (Russian); English transl., Math. USSR-Sb. 66 (1990), no. 1, 265–284. MR 993458, DOI https://doi.org/10.1070/SM1990v066n01ABEH001172 [Isk77]I1 ---, Fano $3$-folds $1$ $($Russian$)$, Izv. Akad. Nauk SSSR Ser. Mat 41 (1977), English transl. in Math. USSR Izv. 11 (1977), 485–527.
- V. A. Iskovskih, Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 3, 516–562, 717 (Russian). MR 463151
- A. N. Tjurin, The intermediate Jacobian of three-dimensional varieties, Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, 1979, pp. 5–57, 239 (loose errata) (Russian). MR 537684 [K92]FA J. Kollár et al., Flips and abundance for algebraic threefolds, vol. 211, Astérisque, 1992.
- Yujiro Kawamata, On the plurigenera of minimal algebraic $3$-folds with $K\mathop {\rlap {\raise 3.25pt\hbox {$∼$}}\rlap {$≈$}\phantom {\approx }}0$, Math. Ann. 275 (1986), no. 4, 539–546. MR 859328, DOI https://doi.org/10.1007/BF01459135
- Yujiro Kawamata, Boundedness of $\mathbf Q$-Fano threefolds, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 439–445. MR 1175897
- János Kollár and Shigefumi Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703. MR 1149195, DOI https://doi.org/10.1090/S0894-0347-1992-1149195-9
- János Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36. MR 986434, DOI https://doi.org/10.1017/S0027763000001240
- János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Henry B. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), no. 6, 1257–1295. MR 568898, DOI https://doi.org/10.2307/2374025
- 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
- Joseph Lipman and Andrew John Sommese, On blowing down projective spaces in singular varieties, J. Reine Angew. Math. 362 (1985), 51–62. MR 809965, DOI https://doi.org/10.1515/crll.1985.362.51
- Tatsuhiro Minagawa, Deformations of weak Fano 3-folds with only terminal singularities, Osaka J. Math. 38 (2001), no. 3, 533–540. MR 1860839
- Tatsuhiro Minagawa, Global smoothing of singular weak Fano 3-folds, J. Math. Soc. Japan 55 (2003), no. 3, 695–711. MR 1978218, DOI https://doi.org/10.2969/jmsj/1191418998
- 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, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI https://doi.org/10.2307/2007050
- Shigeru Mukai, Curves and symmetric spaces. I, Amer. J. Math. 117 (1995), no. 6, 1627–1644. MR 1363081, DOI https://doi.org/10.2307/2375032 [Muk95b]Mu4 ---, New development of the theory of Fano threefolds: Vector bundle method and moduli problem, in Japanese, Sugaku 47 (1995), 125–144, English transl. in Sugaku Expositions 15 (2002), no. 2, 125–150. [Pap01]Papa1 S. Papadakis, Gorenstein rings and Kustin-Miller unprojection, Univ. of Warwick Ph. D. thesis, available at $\text {www.maths.warwick.ac.uk /\~ miles/doctors/Stavros}$, 2001.
- Stavros Argyrios Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geom. 13 (2004), no. 2, 249–268. MR 2047698, DOI https://doi.org/10.1090/S1056-3911-03-00350-3
- Henry C. Pinkham, Factorization of birational maps in dimension $3$, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 343–371. MR 713260
- Stavros Argyrios Papadakis and Miles Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geom. 13 (2004), no. 3, 563–577. MR 2047681, DOI https://doi.org/10.1090/S1056-3911-04-00343-1
- Miles Reid, Canonical $3$-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 273–310. MR 605348
- Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
- Miles Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 5, 695–727. MR 1311389, DOI https://doi.org/10.2977/prims/1195165581
- Takeshi Sano, On classifications of non-Gorenstein ${\bf Q}$-Fano $3$-folds of Fano index $1$, J. Math. Soc. Japan 47 (1995), no. 2, 369–380. MR 1317287, DOI https://doi.org/10.2969/jmsj/04720369
- Takeshi Sano, Classification of non-Gorenstein ${\bf Q}$-Fano $d$-folds of Fano index greater than $d-2$, Nagoya Math. J. 142 (1996), 133–143. MR 1399470, DOI https://doi.org/10.1017/S0027763000005663
- V. V. Šokurov, The existence of a line on Fano varieties, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 922–964, 968 (Russian). MR 548510
- V. V. Šokurov, Smoothness of a general anticanonical divisor on a Fano variety, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 430–441 (Russian). MR 534602 [Taka]G6 H. Takagi, Classification of primary $\mathbb {Q}$-Fano $3$-folds with anti-canonical Du Val K$3$ surfaces. II, preprint.
- Hiromichi Takagi, On classification of $\Bbb Q$-Fano 3-folds of Gorenstein index 2. I, II, Nagoya Math. J. 167 (2002), 117–155, 157–216. MR 1924722, DOI https://doi.org/10.1017/S0027763000025460
- Hiromichi Takagi, On classification of $\Bbb Q$-Fano 3-folds of Gorenstein index 2. I, II, Nagoya Math. J. 167 (2002), 117–155, 157–216. MR 1924722, DOI https://doi.org/10.1017/S0027763000025460
- Kiyohiko Takeuchi, Some birational maps of Fano $3$-folds, Compositio Math. 71 (1989), no. 3, 265–283. MR 1022045
Additional Information
Hiromichi Takagi
Affiliation:
Graduate School of Mathematical Sciences, the University of Tokyo, Tokyo, 153-8914, Japan
Email:
takagi@ms.u-tokyo.ac.jp
Received by editor(s):
June 17, 2004
Received by editor(s) in revised form:
April 6, 2005, April 22, 2005, and May 12, 2005
Published electronically:
June 27, 2005
