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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:
[1] Brenton, L. : On singular complex surfaces with negative canonical bundle, with applications to singular compactification of C2 and 3-dimensional rational singularities . Math. Ann. 248 (1980) 117-134. · Zbl 0407.14013 · doi:10.1007/BF01421952 · eudml:163372
[2] Hidaka, F. and Watanabe, K.-I. : Normal Gorenstein surfaces with ample anti-canonical divisor . Tokyo J. of Math. 4 (1981) 319-330. · Zbl 0496.14023 · doi:10.3836/tjm/1270215157
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[5] Kawamata, Y. , Matsuda, K. and Matsuki, K. : Introduction to the minimal model problem . In Algebraic Geometry Sendai (Ed. Oda Kinokuniya) Adv. Stu. in Pure Math. 10 (North Holland, 1987). · Zbl 0672.14006
[6] Mori, S. : Flip theorem and the existence of minimal models for 3-folds . J. Am. Soc. Sci., 1 (1988) 117-253. · Zbl 0649.14023 · doi:10.2307/1990969
[7] Umezu, Y. : On normal projective surface with trivial dualizing sheaf . Tokyo J. of Math. 4 (1981) 343-354. · Zbl 0496.14025 · doi:10.3836/tjm/1270215159
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