General elephants of three-fold divisorial contractions

Kawakita, Masayuki¹

¹ Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

Journal of the American Mathematical Society, Tome 16 (2003) no. 2, pp. 331-362

Résumé

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 $\le 4$.

Détail

DOI : 10.1090/S0894-0347-02-00416-2

Bibliographie

[1] Artin, M. On the solutions of analytic equations Invent. Math. 1968 277 291

[2] Artin, M. Algebraic approximation of structures over complete local rings Inst. Hautes Études Sci. Publ. Math. 1969 23 58

[3] Corti, Alessio Singularities of linear systems and 3-fold birational geometry 2000 259 312

[4] Cutkosky, Steven Elementary contractions of Gorenstein threefolds Math. Ann. 1988 521 525

[5] Hayakawa, Takayuki Blowing ups of 3-dimensional terminal singularities Publ. Res. Inst. Math. Sci. 1999 515 570

[6] Kawakita, Masayuki Divisorial contractions in dimension three which contract divisors to smooth points Invent. Math. 2001 105 119

[7] Kawamata, Yujiro Divisorial contractions to 3-dimensional terminal quotient singularities 1996 241 246

[8] Kawamata, Yujiro, Matsuda, Katsumi, Matsuki, Kenji Introduction to the minimal model problem 1987 283 360

[9] Kollã¡R, Jã¡Nos, Mori, Shigefumi Classification of three-dimensional flips J. Amer. Math. Soc. 1992 533 703

[10] Kollã¡R, Jã¡Nos, Mori, Shigefumi Birational geometry of algebraic varieties 1998

[11] Kollã¡R, J., Shepherd-Barron, N. I. Threefolds and deformations of surface singularities Invent. Math. 1988 299 338

[12] Markushevich, Dimitri Minimal discrepancy for a terminal cDV singularity is 1 J. Math. Sci. Univ. Tokyo 1996 445 456

[13] Mori, Shigefumi Threefolds whose canonical bundles are not numerically effective Ann. of Math. (2) 1982 133 176

[14] Mori, Shigefumi On 3-dimensional terminal singularities Nagoya Math. J. 1985 43 66

[15] Mori, Shigefumi Flip theorem and the existence of minimal models for 3-folds J. Amer. Math. Soc. 1988 117 253

[16] Reid, Miles Minimal models of canonical 3-folds 1983 131 180

[17] Reid, Miles Young person’s guide to canonical singularities 1987 345 414

[18] Å Okurov, V. V. Smoothness of a general anticanonical divisor on a Fano variety Izv. Akad. Nauk SSSR Ser. Mat. 1979 430 441