Note on Fano 3-folds
type: Item
instance of: scholarly article
title: Note on Fano 3-folds
zbMATH Open document ID: 0790.14037
DOI: 10.1007/BF01264070
author: H. Maeda
published in: Geometriae Dedicata
publication date: 1994-02-09T00:00:00Z
review text: Main theorem: Let \({\mathcal E}\) be an ample vector bundle of rank \(r\) on a Fano 3-fold \(X\) with \(-K_ X=\text{det} {\mathcal E}\). Let \(i(X)\) be the Fano index of \(X\), i.e., the largest integer such that \(K_ X=i(X)H\) for some line bundle \(H\). Then \(i(X)\geq r\) unless \(X\cong\mathbb{P}^ 1\times\mathbb{P}^ 2\) and \({\mathcal E}\cong p^*{\mathcal O}(1)\otimes q^*{\mathcal F}\), where \(p\), \(q\) are the first and second projections and \({\mathcal F}\) is \({\mathcal O}(2)\oplus{\mathcal O}(1)\) or the tangent bundle of \(\mathbb{P}^ 2\). The proof uses Shokurov's existence theorem of lines and the Mori-Mukai classification of Fano 3-folds.
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J60
zbMATH DE Number: 493324
zbMATH Keywords: Fano index
zbMATH Keywords: classification of Fano 3-folds
reviewed by: Takao Fujita
MaRDI profile type: Publication
cites work: Q4027239
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cites work: On uniform vector bundles
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description: scientific article; zbMATH DE number 493324