# zbMATH — the first resource for mathematics

General elephants of $${\mathbb{Q}}$$-Fano 3-folds. (English) Zbl 0813.14028
A general elephant is, as introduced by M. Reid, a general member of the anti-canonical system of a 3-fold. The author studies the important and interesting problem under what circumstances a general elephant has only Du Val singularities.
V. V. Shokurov [Math. USSR, Izv. 14, 395-405 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 430-441 (1979; Zbl 0407.14017)] has shown the smoothness of a general elephant on a smooth Fano 3-fold. This result enters essentially into V. A. Iskovskih’s famous classification of Fano 3-folds. There exists a generalization due to M. Reid [“Projective morphisms according to Kawamata” (Preprint, Warwick 1983)] stating that a general elephant on a Gorenstein 3-fold $$X$$ with canonical singularities and with big and nef $$K^{-1}_ X$$ has only Du Val singularities (being the same as canonical surface singularities). – In this paper, the author considers the case of $$\mathbb{Q}$$-Fano 3-folds. Such a variety is by definition $$\mathbb{Q}$$-factorial, has only terminal singularities, their Picard number is one and the anticanonical divisor is ample.
The main theorems will not state the expected statement about the singularities of general elephants. The author proves that this will be true after a birational transformation. More precisely, he obtains the following theorem:
If $$X$$ is a $$\mathbb{Q}$$-Fano 3-fold whose anti-canonical map has three- dimensional image, then $$X$$ is birationally equivalent to a $$\mathbb{Q}$$-Fano 3-fold $$X'$$ such that a general member of its anti-canonical system has only Du Val singularities.
He also proves under the same assumptions on $$X$$ the existence of a Fano variety $$Z$$ which is birationally equivalent to $$X$$ and has at most canonical Gorenstein singularities and a base-point free anti-canonical system.
The main technique to obtain his results is Mori’s minimal model program. At the beginning of the paper the author explains shortly the main steps in the minimal model program. Then he introduces two new categories of 3- folds where, as he proves, the minimal model program also works. To understand his new “canonical” category, for example, one should remember the definition of “log-canonical”.
Let $$X$$ be a 3-fold and $$B_ i$$ distinct Weil-divisors. Consider a $$\mathbb{Q}$$-Cartier divisor $$K + \sum b_ iB_ i = K + B$$ with $$0 \leq b_ i \leq 1$$. If $$f : \widetilde X \to X$$ is a “good” resolution of singularities then we define $$\widetilde B : = \sum_ i b_ i \widetilde B_ i + \sum_ j E_ j$$, where $$\widetilde B_ i$$ is the strict transform of $$B_ i$$ and the $$E_ j$$ are the exceptional divisors. By definition, the pair $$(X,B)$$ has logcanonical singularities, if we have $$K_{\widetilde X} + \widetilde B = f^* (K + B) + \sum \gamma_ j E_ j$$ with $$\gamma_ j \geq 0$$.
It is important in the minimal model program to stay in the same category of singularities after divisorial contractions for example. If one of the components $$B_ i$$ will be contracted, a sufficient condition for the result to be again log-canonical is $$b_ i \leq 1$$. But the author need not consider this property, because he replaces $$K + \sum b_ i B_ i$$ with $$K + \sum l_ i L_ i$$ using movable linear systems $$L_ i$$. In the definition of log-canonical singularities he takes $$\sum l_ i \widetilde L_ i$$ instead of $$\widetilde B$$ to obtain his new category.
The birationally equivalent varieties mentioned in the theorem are constructed by applying the minimal model program in his new “terminal” and “canonical” categories.

##### MSC:
 14J30 $$3$$-folds 14J45 Fano varieties
Full Text:
##### References:
 [1] V. Alexeev , Ample Weil divisors on K3 surfaces with Du Val singularities , Duke Math. J. 64 (1991), 617-624. · Zbl 0766.14030 · doi:10.1215/S0012-7094-91-06429-X [2] M. Artin , On isolated rational singularities of surfaces , Am. J. Math. 88 (1966), 129-137. · Zbl 0142.18602 · doi:10.2307/2373050 [3] A. Conte - J.P. Murre , Algebraic varieties of dimension three whose hyperplane sections are Enriques surfaces , Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser. 12 (1985), 43-80. · Zbl 0612.14041 · numdam:ASNSP_1985_4_12_1_43_0 · eudml:83952 [4] J.-L. Colliot-Thélène , Arithmétique des variétés rationelles et problèmes birationelles , In: ”Proc. Int. Conf. Math.”, 1986, 641-653. · Zbl 0698.14060 [5] A. Corti , Thesis , in preparation. [6] G. Fano , Sulle varietà algebriche a tre dimensioni le cui sezioni iperpiane sono superficie di genere zero e bigenere uno , Memoire società dei XL, 24 (1938), 41-66. · Zbl 0022.07702 [7] A.R. Fletcher , Letter to V. A. Iskovskih , 1987. [8] A.R. Fletcher , Working with weighted complete intersections , Max-Plank Institute preprint, 1989. · Zbl 0960.14027 [9] V.A. Iskovskih , Fano threefolds I, II , Izv. Akad. Nauk SSSR, 41 (1977), 516-562, resp. 42 (1978), 506-549. · Zbl 0407.14016 [10] V.A. Iskovskih , Lectures on Algebraic Threefolds. Fano Varieties , Publ. by Moscow University, 1988. [11] Y. Kawamata , Boundedness of Q-Fano threefolds , preprint, 1989. [12] Y. Kawamata , Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces , Ann. of Math. 127 (1988), 93-163. · Zbl 0651.14005 · doi:10.2307/1971417 [13] Y. Kawamata , The minimal discrepancy coefficients of terminal singularities in dimension 3 . Appendix to [Sh3]. [14] Y. Kawamata , Termination of log flips for algebraic 3-folds , preprint, 1991. · Zbl 0814.14016 · doi:10.1142/S0129167X92000308 [15] Y. Kawamata , K. Matsuda and K. Matsuki , Introduction to the Minimal Model Program , In: ” Algebraic Geometry , Sendai”, Adv. Stud. Pure Math. 10 (1987), 283-360. · Zbl 0672.14006 [16] K. Kodaira , On Compact Analytic Surfaces, I. II, III , In ” Collected works ”, v. 3, Iwanami Shoten, Publishers and Princeton University Press, 1975. · Zbl 0315.01027 · doi:10.1515/9781400869879 [17] J. Kollár , Flops , Nagoya Math. J., 113 (1989), 14-36. · Zbl 0645.14004 · doi:10.1017/S0027763000001240 [18] J. Kollár , Flips, flops, minimal models etc , preprint, 1990. · Zbl 0755.14003 [19] Yu. I. Manin , Cubic forms . · Zbl 0255.14002 [20] S. Mori , Threefolds whose canonical bundles are not numerically effective , Ann. of Math. 116 (1982), 133-176. · Zbl 0557.14021 · doi:10.2307/2007050 [21] S. Mori , Classification of Higher Dimensional Varieties , In: ” Algebraic geometry Bowdoin 1985 ”, Proc. Symp. Pure Math. 46 (1987), 269-332. · Zbl 0656.14022 [22] S. Mori , Lectures on Fano 3-folds , University of Utah, 1988. [23] V.V. Nikulin , Linear systems on singular K3 surfaces , preprint, 1990. · Zbl 0785.14021 [24] M. Reid , Projective morphisms according to Kawamata , preprint, 1983. [25] M. Reid , Young Person’s Guide to Canonical Singularities , In: ” Algebraic geometry Bowdoin 1985 ”, Proc. Symp. Pure Math. 46 AMS, 1987, 345-416. · Zbl 0634.14003 [26] M. Reid , Birational geometry of 3-folds according to Sarkisov , unpublished lecture notes, 1991. [27] V.V. Shokurov , Smoothness of a general anticanonical divisor on a Fano variety , Izv. Akad. Nauk SSSR, 14 (1980), 395-405. · Zbl 0429.14012 · doi:10.1070/IM1980v014n02ABEH001123 [28] V.V. Shokurov , The nonvanishing theorem , Izv. Akad. Nauk SSSR, 49 (1985), 635-651. · Zbl 0605.14006 · doi:10.1070/IM1986v026n03ABEH001160 [29] V.V. Shokurov , 3-fold log flips , preprint, 1991 (translated by M. Reid). · Zbl 0828.14027 [30] T. Urabe , Fixed components of linear systems on K3 surfaces , preprint, 1990. [31] P.M.H. Wilson , Varieties , Bull. London Math. Soc., 19 (1987), 1-48. · Zbl 0612.14033 · doi:10.1112/blms/19.1.1 [32] P.M.H. Wilson , Toward a birational classification of algebraic varieties , Bull. London Math. Soc., 19 (1987), 1-48. · Zbl 0612.14033 · doi:10.1112/blms/19.1.1 [33] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.