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Classification of extremal contractions
from smooth fourfolds of $(3,1)$-type

Author: Hiromichi Takagi
Journal: Proc. Amer. Math. Soc. 127 (1999), 315-321
MSC (1991): Primary 14E30; Secondary 14J35
MathSciNet review: 1637436
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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.

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  • [A] T. Ando, On extremal rays of the higher dimensional varieties, Invent. Math. 81 (1985), 347-357. MR 87g:14045
  • [A-W1] M. Andreatta and J. Wi\'{s}niewski, A note on nonvanishing and applications, Duke Math. J. 72 (1993), 739-755. MR 95c:14007
  • [A-W2] -, On contractions of smooth varieties, J. Algebraic Geom. 7 (1998), 253-312. CMP 98:12
  • [A-W3] -, A view on contractions of higher dimensional varieties, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 153-183. CMP 98:07
  • [Be1] M. Beltrametti, On $d$-folds whose canonical bundle is not numerically effective,According to Mori and Kawamata, Ann. Mat. Pura. Appl. 147 (1987), 151-172. MR 89c:14063
  • [Be2] -, Contraction of non numerically effective extremal rays in dimension $4$, Teubner-Texte Math. 92 (1986), 24-37. MR 89c:14064
  • [F0] T. Fujita, Classification theories of polarized varieties, London Math. Soc. Lecture Note Ser. 155 (1990). MR 93e:14009
  • [F1] -, On singular del Pezzo varieties, Lecture Notes in Math., vol. 1417, Springer-Verlag, 1990, p. 117-128. MR 91b:14057
  • [H] R. Hartshorne, Algebraic Geometry, GTM 52 (1977). MR 57:3116
  • [Kac] Y. Kachi, Extremal contractions from 4-dimensional manifolds to 3-folds, Ann. di Pisa (4) 24 (1997), 63-131. CMP 98:03
  • [Kaw1] Y. Kawamata, The cone of curves of algebraic varieties, Ann. of Math. 119 (1984), 603-633. MR 86c:14013b
  • [Kaw2] -, Small contractions of four dimensional algebraic manifolds, Math. Ann. 284 (1989), 595-600. MR 91e:14039
  • [KMM] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Adv. St. Pure Math. 10 (1987), 287-360. MR 89e:14015
  • [Mo1] S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 110 (1979), 593-606. MR 81j:14010
  • [Mo2] -, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176. MR 84e:14032
  • [N] S. Nakano, On the inverse of monoidal transformations, Publ. RIMS Kyoto Univ. 6 (1971), 483-502. MR 45:3778; MR 45:3779; MR 45:3780
  • [R1] M.Reid, Canonical 3-folds, Journées de Géométrie Algébrique d'Angers, Sijthoff and Noordhoff, Alphen, 1980, p. 273-310. MR 82i:14025
  • [R2] -, Minimal models of canonical 3-folds, Adv. St. Pure Math. 1 (1983), 131-180. MR 86a:14010
  • [R3] -, Nonnormal del Pezzo surfaces, Publ. RIMS Kyoto Univ. 30 (1994), 695-727. MR 96a:14042
  • [S] M.Schlessinger, Rigidity of quotient singularities, Invent.Math. 14 (1971), 17-26. MR 45:1912
  • [W1] P. M. H. Wilson, The Kähler cone on Calabi Yau threefolds (and Erratum), Invent. Math. 107 (114) (1992 (1993)), 561-583 (231-233). MR 93a:14037; MR 94i:14042
  • [W2] -, Symplectic Deformations of Calabi-Yau threefolds, J. Diff. Geom. 45 (1997), 611-637. CMP 98:02

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Additional Information

Hiromichi Takagi
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-0041, Japan

Keywords: Extremal ray, contraction morphism
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
Article copyright: © Copyright 1999 American Mathematical Society

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