## General elephants of three-fold divisorial contractions

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- by Masayuki Kawakita PDF
- J. Amer. Math. Soc.
**16**(2003), 331-362 Request permission

## Abstract:

We treat three-fold divisorial contractions whose exceptional divisors contract to Gorenstein points. We prove that a general element in the anti-canonical system around the exceptional divisor has at worst Du Val singularities. As application to classification, we describe divisorial contractions to compound $A_{n}$ points, and moreover, we deduce that any divisorial contraction to a compound $D_{n}$ or $E_{n}$ point has discrepancy $\le 4$.## References

- M. Artin,
*On the solutions of analytic equations*, Invent. Math.**5**(1968), 277–291. MR**232018**, DOI 10.1007/BF01389777 - M. Artin,
*Algebraic approximation of structures over complete local rings*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 23–58. MR**268188**, DOI 10.1007/BF02684596 - Alessio Corti,
*Singularities of linear systems and $3$-fold birational geometry*, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 259–312. MR**1798984**
CM00A. Corti and M. Mella, Birational geometry of terminal quartic $3$-folds I, preprint (2000).
- Steven Cutkosky,
*Elementary contractions of Gorenstein threefolds*, Math. Ann.**280**(1988), no. 3, 521–525. MR**936328**, DOI 10.1007/BF01456342 - Takayuki Hayakawa,
*Blowing ups of $3$-dimensional terminal singularities*, Publ. Res. Inst. Math. Sci.**35**(1999), no. 3, 515–570. MR**1710753**, DOI 10.2977/prims/1195143612 - Masayuki Kawakita,
*Divisorial contractions in dimension three which contract divisors to smooth points*, Invent. Math.**145**(2001), no. 1, 105–119. MR**1839287**, DOI 10.1007/s002220100144
Ka02—, Divisorial contractions in dimension three which contract divisors to compound $A_1$ points, Compos. Math. - Yujiro Kawamata,
*Divisorial contractions to $3$-dimensional terminal quotient singularities*, Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 241–246. MR**1463182** - 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 - János Kollár and Shigefumi Mori,
*Classification of three-dimensional flips*, J. Amer. Math. Soc.**5**(1992), no. 3, 533–703. MR**1149195**, DOI 10.1090/S0894-0347-1992-1149195-9 - János Kollár and Shigefumi Mori,
*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR**1658959**, DOI 10.1017/CBO9780511662560 - J. Kollár and N. I. Shepherd-Barron,
*Threefolds and deformations of surface singularities*, Invent. Math.**91**(1988), no. 2, 299–338. MR**922803**, DOI 10.1007/BF01389370 - Dimitri Markushevich,
*Minimal discrepancy for a terminal cDV singularity is $1$*, J. Math. Sci. Univ. Tokyo**3**(1996), no. 2, 445–456. MR**1424437** - 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 - Shigefumi Mori,
*On $3$-dimensional terminal singularities*, Nagoya Math. J.**98**(1985), 43–66. MR**792770**, DOI 10.1017/S0027763000021358 - Shigefumi Mori,
*Flip theorem and the existence of minimal models for $3$-folds*, J. Amer. Math. Soc.**1**(1988), no. 1, 117–253. MR**924704**, DOI 10.1090/S0894-0347-1988-0924704-X - 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
Re83p—, Projective morphisms according to Kawamata, preprint (1983).
- 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** - 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**
Ta00H. Takagi, On classification of ${\mathbb Q}$-Fano $3$-folds of Gorenstein index $2$ II, Nagoya Math. J.

**133**(2002), 95-116.

**167**(2002), 157-216.

## Additional Information

**Masayuki Kawakita**- Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
- MR Author ID: 680001
- Email: kawakita@ms.u-tokyo.ac.jp
- Received by editor(s): October 22, 2001
- Received by editor(s) in revised form: September 4, 2002
- Published electronically: December 2, 2002
- © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 331-362 - MSC (2000): Primary 14E05, 14E30
- DOI: https://doi.org/10.1090/S0894-0347-02-00416-2
- MathSciNet review: 1949163