Kyoto Journal of Mathematics

Threefold extremal contractions of type (IA)

Shigefumi Mori and Yuri Prokhorov

Full-text: Open access


Let (X,C) be a germ of a threefold X with terminal singularities along an irreducible reduced complete curve C with a contraction f:(X,C)(Z,o) such that C=f1(o)red and KX is ample. Assume that a general member F|KX| meets C only at one point P, and furthermore assume that (F,P) is Du Val of type A if index (X,P)=4. We classify all such germs in terms of a general member H|OX| containing C.

Article information

Kyoto J. Math., Volume 51, Number 2 (2011), 393-438.

First available in Project Euclid: 22 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J30: $3$-folds [See also 32Q25] 14E 14E30: Minimal model program (Mori theory, extremal rays)


Mori, Shigefumi; Prokhorov, Yuri. Threefold extremal contractions of type (IA). Kyoto J. Math. 51 (2011), no. 2, 393--438. doi:10.1215/21562261-1214393.

Export citation


  • [Art] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291.
  • [Bin] J. Bingener, On the existence of analytic contractions, Invent. Math. 64 (1981), 25–67.
  • [Fuj] A. Fujiki, Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79), 1–52.
  • [Gro] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, with an exposé by M. Raynaud, Séminaire de Géométrie Algébrique du Bois-Marie 1962 (SGA 2), Adv. Stud. Pure Math. 2, North-Holland, Amsterdam, 1968.
  • [Kaw] Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2), 127 (1988), 93–163.
  • [Kol] J. Kollár, ed., Flips and abundance for algebraic threefolds (Salt Lake City, 1991), Astérisque 211, Soc. Math. France-Montrouge, 1992.
  • [KM] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), 533–703.
  • [KSB] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), 299–338.
  • [LW] E. Looijenga and J. Wahl, Quadratic functions and smoothing surface singularities, Topology 25 (1986), 261–291.
  • [Mor1] S. Mori, On 3-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66.
  • [Mor2] S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), 117–253.
  • [Mor3] S. Mori, “On semistable extremal neighborhoods” in Higher Dimensional Birational Geometry (Kyoto, 1997), Adv. Stud. Pure Math. 35, Math. Soc. Japan, Tokyo, 2002, 157–184.
  • [MP1] S. Mori and Y. Prokhorov, On Q-conic bundles, Publ. Res. Inst. Math. Sci. 44 (2008), 315–369.
  • [MP2] S. Mori and Y. Prokhorov, On Q-conic bundles, III, Publ. Res. Inst. Math. Sci. 45 (2009), 787–810.
  • [Nak] N. Nakayama, “The lower semicontinuity of the plurigenera of complex varieties” in Algebraic Geometry (Sendai, Japan, 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam, 1987, 551–590.
  • [Pro1] Y. Prokhorov, On the complementability of the canonical divisor for Mori fibrations on conics, Sbornik. Math. 188 (1997), 1665–1685.
  • [Pro2] Y. Prokhorov, Lectures on complements on log surfaces, MSJ Memoirs 10, Math. Soc. Japan, Tokyo, 2001.
  • [Rei1] M. Reid, “Minimal models of canonical 3-folds” in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983, 131–180.
  • [Rei2] M. Reid, “Young person’s guide to canonical singularities” in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math. 46, Amer. Math. Soc., Providence, 1987, 345–414.
  • [SB] N. I. Shepherd-Barron, “Degenerations with numerically effective canonical divisor” in The Birational Geometry of Degenerations (Cambridge, Mass., 1981), Progr. Math. 29, Birkhäuser, Boston, 1983, 33–84.
  • [Sho] V. V. Shokurov, 3-fold log flips, Russ. Acad. Sci. Izv. Math. 40 (1993), 95–202.
  • [Ste] J. Stevens, On canonical singularities as total spaces of deformations, Abh. Math. Sem. Univ. Hamburg 58 (1988), 275–283.
  • [Tzi1] N. Tziolas, Three dimensional divisorial extremal neighborhoods, Math. Ann. 333 (2005), 315–354.
  • [Tzi2] N. Tziolas, ℚ-Gorenstein deformations of nonnormal surfaces, Amer. J. Math. 131 (2009), 171–193.