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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).

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).

Reviewer: N.Mohan Kumar (Bombay)

### 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 |

### 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 |

[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 |

[3] | Kawamata, Y. : The cone of curves of algebraic varieties . Ann. of Math. 119 (1984) 603-633. · Zbl 0544.14009 |

[4] | Kawamata, Y. : Crepant blowing ups of 3-dimensional singularities and its application to degenerations of surfaces . Ann. of Math. 127 (1988) 93-163. · Zbl 0651.14005 |

[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 |

[7] | Umezu, Y. : On normal projective surface with trivial dualizing sheaf . Tokyo J. of Math. 4 (1981) 343-354. · Zbl 0496.14025 |

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