Classification of extremal contractions from smooth fourfolds of (3,1)-type
HTML articles powered by AMS MathViewer
- by Hiromichi Takagi PDF
- Proc. Amer. Math. Soc. 127 (1999), 315-321 Request permission
Abstract:
We investigate divisorial contractions of extremal rays from smooth fourfolds. When the exceptional divisor is contracted to a curve, we prove that the divisor is a $\mathbb {P}^{2}$-bundle or quadric bundle over a smooth curve and the contraction is the blowing up along the curve. Furthermore we determine the local analytic structure of the contraction.References
- Tetsuya Ando, On extremal rays of the higher-dimensional varieties, Invent. Math. 81 (1985), no. 2, 347–357. MR 799271, DOI 10.1007/BF01389057
- M. Andreatta and J. A. Wiśniewski, A note on nonvanishing and applications, Duke Math. J. 72 (1993), no. 3, 739–755. MR 1253623, DOI 10.1215/S0012-7094-93-07228-6
- —, On contractions of smooth varieties, J. Algebraic Geom. 7 (1998), 253–312.
- —, A view on contractions of higher dimensional varieties, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 153–183.
- Mauro Beltrametti, On $d$-folds whose canonical bundle is not numerically effective, according to Mori and Kawamata, Ann. Mat. Pura Appl. (4) 147 (1987), 151–172 (English, with Italian summary). MR 916706, DOI 10.1007/BF01762415
- Mauro Beltrametti, Contractions of nonnumerically effective extremal rays in dimension $4$, Proceedings of the conference on algebraic geometry (Berlin, 1985) Teubner-Texte Math., vol. 92, Teubner, Leipzig, 1986, pp. 24–37. MR 922898
- Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR 1162108, DOI 10.1017/CBO9780511662638
- 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 10.1007/BFb0083337
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Y. Kachi, Extremal contractions from 4-dimensional manifolds to 3-folds, Ann. di Pisa (4) 24 (1997), 63–131.
- Yujiro Kawamata, Elementary contractions of algebraic $3$-folds, Ann. of Math. (2) 119 (1984), no. 1, 95–110. MR 736561, DOI 10.2307/2006964
- Yujiro Kawamata, Small contractions of four-dimensional algebraic manifolds, Math. Ann. 284 (1989), no. 4, 595–600. MR 1006374, DOI 10.1007/BF01443353
- 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 10.2969/aspm/01010283
- Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
- Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. MR 662120, DOI 10.2307/2007050
- Shigeo Nakano, On the inverse of monoidal transformation, Publ. Res. Inst. Math. Sci. 6 (1970/71), 483–502. MR 0294710, DOI 10.2977/prims/1195193917
- 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, Minimal models of canonical $3$-folds, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 131–180. MR 715649, DOI 10.2969/aspm/00110131
- Miles Reid, Nonnormal del Pezzo surfaces, Publ. Res. Inst. Math. Sci. 30 (1994), no. 5, 695–727. MR 1311389, DOI 10.2977/prims/1195165581
- Michael Schlessinger, Rigidity of quotient singularities, Invent. Math. 14 (1971), 17–26. MR 292830, DOI 10.1007/BF01418741
- P. M. H. Wilson, The Kähler cone on Calabi-Yau threefolds, Invent. Math. 107 (1992), no. 3, 561–583. MR 1150602, DOI 10.1007/BF01231902
- —, Symplectic Deformations of Calabi-Yau threefolds, J. Diff. Geom. 45 (1997), 611–637.
Additional Information
- Hiromichi Takagi
- Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-0041, Japan
- Email: htakagi@ms.u-tokyo.ac.jp
- Received by editor(s): February 12, 1997
- Received by editor(s) in revised form: April 24, 1997
- Additional Notes: The author is a Research Fellow of the Japan Society for the Promotion of Science
- Communicated by: Ron Donagi
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 315-321
- MSC (1991): Primary 14E30; Secondary 14J35
- DOI: https://doi.org/10.1090/S0002-9939-99-05114-X
- MathSciNet review: 1637436