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Threefold extremal contractions of type (IA). (English) Zbl 1230.14017

One of the fundamental problems of three dimensional birational geometry is the explicit classification of divisorial contractions, flips and Mori fiber spaces (conic bundles and Del Pezzo fibrations). The existence of flips was established by S. Mori in [J. Am. Math. Soc. 1, No. 1, 117–253 (1988; Zbl 0649.14023)] and their classification by S. Mori and J. Kollár [J. Am. Math. Soc. 5, No. 3, 533–703 (1992; Zbl 0773.14004)]. Divisorial contractions with at most one dimensional fibers, flipping contractions and conic bundles have many similarities in the sense that locally over the base they may be viewed as a morphism \(f : (X,C) \rightarrow (Y,Q)\), where \((X,C)\) is the germ of a 3-fold \(X\) with \(\mathbb{Q}\)-factorial terminal singularities along a proper curve \(C\) such that \(-K_X\) is \(f\)-ample and \(f^{-1}(Q)_{red}=C\). Such a map is called an extremal curve germ. If \(f\) is birational then \(f\) is either a divisorial contraction or a flip and is called an extremal neighborhood. Otherwise it is a conic bundle. In all cases, \(f\) is a one parameter \(\mathbb{Q}\)-Gorenstein smoothing of the map \(f_0 : H \rightarrow T\), where \(H \in |\mathcal{O}_X|\) is the general hyperplace section of \(X\) containing \(C\) and \(T=f(H)\). This is the approach originally taken by S. Mori and J. Kollár in their classification of flips. Therefore a classification of 3-fold extremal curve germs \(f : (X,C) \rightarrow (Y,P)\) can be obtained by classifying the general members \(H \in |\mathcal{O}_X|\) containing \(C\) and their deformations.
In this paper the authors study extremal curve germs \(f : (X,C) \rightarrow (Y,Q)\) in the case that \(X\) has singularities of type (IA) (according to the classification of 3-fold terminal singularities and the notation [J. Am. Math. Soc. 1, No. 1, 117–253 (1988; Zbl 0649.14023)]). In particular, let \(H \in |\mathcal{O}_X|\) be the general member. The authors distinguish cases with respect to the singularities of \(X\) when \(H\) is normal or not. Moreover, they show that in all cases \(H^{\prime}\) has rational singularities (where \(H^{\prime}\) is the normalization of \(H\)) and they classify the dual graphs \(\Delta(H^{\prime},C^{\prime})\), where \(C^{\prime}\) is the inverse image of \(C\) in the normalization of \(H\).

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14J30 \(3\)-folds
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References:

[1] M. Artin, On the solutions of analytic equations , Invent. Math. 5 (1968), 277-291. · Zbl 0172.05301 · doi:10.1007/BF01389777
[2] J. Bingener, On the existence of analytic contractions , Invent. Math. 64 (1981), 25-67. · Zbl 0509.32004 · doi:10.1007/BF01393933
[3] A. Fujiki, Closedness of the Douady spaces of compact Kähler spaces , Publ. Res. Inst. Math. Sci. 14 (1978/79), 1-52. · Zbl 0409.32016 · doi:10.2977/prims/1195189279
[4] 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. · Zbl 0197.47202
[5] 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. JSTOR: · Zbl 0651.14005 · doi:10.2307/1971417
[6] J. Kollár, ed., Flips and abundance for algebraic threefolds (Salt Lake City, 1991) , Astérisque 211 , Soc. Math. France-Montrouge, 1992.
[7] J. Kollár and S. Mori, Classification of three-dimensional flips , J. Amer. Math. Soc. 5 (1992), 533-703. · Zbl 0773.14004 · doi:10.2307/2152704
[8] J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities , Invent. Math. 91 (1988), 299-338. · Zbl 0642.14008 · doi:10.1007/BF01389370
[9] E. Looijenga and J. Wahl, Quadratic functions and smoothing surface singularities , Topology 25 (1986), 261-291. · Zbl 0615.32014 · doi:10.1016/0040-9383(86)90044-3
[10] S. Mori, On 3-dimensional terminal singularities , Nagoya Math. J. 98 (1985), 43-66. · Zbl 0589.14005
[11] S. Mori, Flip theorem and the existence of minimal models for 3-folds , J. Amer. Math. Soc. 1 (1988), 117-253. · Zbl 0649.14023 · doi:10.2307/1990969
[12] 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. · Zbl 1066.14018
[13] S. Mori and Y. Prokhorov, On Q - conic bundles , Publ. Res. Inst. Math. Sci. 44 (2008), 315-369. · Zbl 1151.14029 · doi:10.2977/prims/1210167329
[14] S. Mori and Y. Prokhorov, On Q - conic bundles, III , Publ. Res. Inst. Math. Sci. 45 (2009), 787-810. · Zbl 1182.14040 · doi:10.2977/prims/1249478965
[15] 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. · Zbl 0649.14003
[16] Y. Prokhorov, On the complementability of the canonical divisor for Mori fibrations on conics , Sbornik. Math. 188 (1997), 1665-1685. · Zbl 0921.14025
[17] Y. Prokhorov, Lectures on complements on log surfaces , MSJ Memoirs 10 , Math. Soc. Japan, Tokyo, 2001. · Zbl 1037.14003
[18] 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. · Zbl 0558.14028
[19] 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. · Zbl 0634.14003
[20] 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. · Zbl 0506.14028
[21] V. V. Shokurov, 3-fold log flips , Russ. Acad. Sci. Izv. Math. 40 (1993), 95-202. · Zbl 0828.14027
[22] J. Stevens, On canonical singularities as total spaces of deformations , Abh. Math. Sem. Univ. Hamburg 58 (1988), 275-283. · Zbl 0703.14003 · doi:10.1007/BF02941384
[23] N. Tziolas, Three dimensional divisorial extremal neighborhoods , Math. Ann. 333 (2005), 315-354. · Zbl 1083.14014 · doi:10.1007/s00208-005-0676-9
[24] N. Tziolas, \Bbb Q- Gorenstein deformations of nonnormal surfaces , Amer. J. Math. 131 (2009), 171-193. · Zbl 1162.14005 · doi:10.1353/ajm.0.0039
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