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Quasi-Gorenstein Fano 3-folds with isolated non-rational loci. (English) Zbl 0738.14025
From the introduction: …A quasi-Gorenstein Fano $$n$$-fold is, by definition, an $$n$$-dimensional projective variety $$X$$ over $$\mathbb{C}$$ such that the anti-canonical divisor $$-K_ X$$ is an ample Cartier divisor. Put $$\Sigma_ X=\{x\in X\mid x$$ is not a rational singularity on $$X$$}. Then $$\Sigma_ X$$ is a closed subset of $$X$$ of codimension at least two. L. Brenton [Math. Ann. 248, 117-124 (1980; Zbl 0407.14013)], F. Hidaka and K. Watanabe [Tokyo J. Math. 4, 319-330 (1981; Zbl 0496.14023)] determined all quasi-Gorenstein Fano surfaces. As they show, both cases $$\Sigma_ X=\emptyset$$ and $$\Sigma_ X\neq\emptyset$$ occur.
We treat the latter case $$\Sigma_ X\neq\emptyset$$. We try to clarify the structure of a quasi-Gorenstein Fano $$n$$-fold with $$\dim(\Sigma_ X)=0$$. If we assume the minimal model conjecture, it turns out to have the structure of a projective cone defined by an ample invertible sheaf $${\mathcal L}$$ on a normal $$(n-1)$$-fold $$S$$ staisfying $${\mathcal O}(K_ S)\cong{\mathcal O}_ S$$ with at worst rational singularities on $$S$$. Here the projective cone defined by $${\mathcal L}$$ on $$S$$ means the normal projective variety obtained by contracting the negative section of $$\mathbb{P}({\mathcal O}_ S\oplus{\mathcal L})$$. Since the minimal model conjecture is known to hold for surfaces, a quasi-Gorenstein Fano surface with $$\Sigma_ X\neq\emptyset$$ (necessarily $$\dim(\Sigma_ X)=0$$) is the projective cone defined by an ample invertible sheaf $${\mathcal L}$$ on an elliptic curve; this result is found in the papers by L. Brenton (loc. cit.) and F. Hidaka and K. Watanabe (loc. cit). — For three-folds, the minimal model conjecture was proved by S. Mori [J. Am. Math. Soc. 1, No. 1, 117-253 (1988; Zbl 0649.14023)]. Therefore we have that a quasi-Gorenstein Fano 3-fold with $$\dim(\Sigma_ X)=0$$ is the projective cone defined by an ample invertible sheaf $${\mathcal L}$$ on either an Abelian surface or a normal $$K3$$-surface (i.e. a normal surface with trivial canonical sheaf whose minimal resolution is a $$K3$$-surface).

##### MSC:
 14J45 Fano varieties 14J30 $$3$$-folds 14B05 Singularities in algebraic geometry 14J17 Singularities of surfaces or higher-dimensional varieties 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14E30 Minimal model program (Mori theory, extremal rays) 14J10 Families, moduli, classification: algebraic theory
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##### References:
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