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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
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References:
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