A type of the Lefschetz hyperplane section theorem on \({\mathbb{Q}}\)-Fano 3-folds with Picard number one and \({\frac{1}{2}(1,1,1)}\)-singularities
type: Item
instance of: scholarly article
title: A type of the Lefschetz hyperplane section theorem on \({\mathbb{Q}}\)-Fano 3-folds with Picard number one and \({\frac{1}{2}(1,1,1)}\)-singularities
zbMATH Open document ID: 1253.14039
author: Nam-Hoon Lee
published in: Geometriae Dedicata
publication date: 2012-07-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.5946
review text: Let \(X\) be a projective \(3\)-fold with cyclic singularities of type \(\frac{1}{2}(1,1,1)\) and with Picard number \(1\). Assume further that the class group of \(X\) is generated by one element \(H\), that \({-}2K_X\) is very ample, and that the linear system \(|{-}K_X|\) has a member which is smooth on the smooth locus of \(X\), and has singularities of type \(\frac{1}{2}(1,1)\) elsewhere. If \(D\sim nH\) is a Weil divisor which has singularities of type \(\frac{1}{2}(1,1)\) along the singular locus of \(X\) and is otherwise smooth, then the main result of the paper under review is that the map \(\mathrm{Cl}(X)\to \mathrm{Cl}(D)/(\mathrm{torsion})\) is injective and has a torsion-free cokernel. Then the author gives a new example of a Calabi-Yau \(3\)-fold with \(\text{Pic}(X)=\mathbb{Z}H\), where \(H^3=8\), \(c_2(X)\cdot H=44\), and whose topological Euler characteristic is \(-88\).
reviewed by: Vladimir Lazic
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 6062609
zbMATH Keywords: Fano Q-folds
zbMATH Keywords: Leftschetz hyperplane section theorem
zbMATH Keywords: Calabi-Yau 3-folds
MaRDI profile type: Publication
OpenAlex ID: W1970159171
arXiv ID: 1107.5946
cites work: Q4695251
cites work: Primitive Calabi-Yau threefolds
cites work: Calabi-Yau coverings over some singular varieties and new Calabi-Yau 3-folds with Picard number one
cites work: CALABI–YAU CONSTRUCTION BY SMOOTHING NORMAL CROSSING VARIETIES
cites work: Classification of non-Gorenstein Q-Fano d-folds of Fano index greater than d − 2
cites work: On classification of non-Gorenstein \(\mathbb{Q}\)-Fano 3-folds of Fano index 1
cites work: On classification of ℚ-fano 3-folds of Gorenstein index 2. I
cites work: On classification of ℚ-fano 3-folds of Gorenstein index 2. II
cites work: Q3438276
DOI: 10.1007/S10711-011-9644-6
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description: scientific article; zbMATH DE number 6062609