Fano 3-folds with divisible anticanonical class
type: Item
instance of: scholarly article
title: Fano 3-folds with divisible anticanonical class
zbMATH Open document ID: 1119.14036
author: Kaori Suzuki
author: Gavin Brown
published in: Manuscripta Mathematica
publication date: 2007-06-07T00:00:00Z
review text: Let \(X\) be a Fano \(3\)-fold, i.e. a normal projective \(3\)-fold with \(\mathbb Q\)-factorial singularities such that \(-K_X\) is ample and \(\rho(X)=1\). The index \(f\) of \(X\) is the biggest integer \(f>0\) such that \(-K_X=fA\) for some Weil divisor \(A\). The divisor \(A\) is a primitive ample divisor and there is a corresponding graded ring \(R(X,A)=\bigoplus _{n\geq 0} H^0(X,\mathcal O _X(nA))\). In the paper under review, it is shown that there are at most 1,492 power series corresponding to the Hilbert series of \(R(X,A)\) where \(X\) is a Fano \(3\)-fold of index \(2\) with \(-K_X=2A\). From this classification it then follows that \((-K_X)^3\geq 8/165\) and that if \(X\) is a Fano 3-fold of index at least \(2\), then \(H^0(X,-K_X)\neq 0\). The case in which \(f\geq 3\) had been previously established in ``Computing Fano 3-folds of index \(\geq 3\)'' [to appear in Jpn. J. Indust. Appl. Math. 11 (2007)].
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14Q15
zbMATH DE Number: 5162386
zbMATH Keywords: Fano 3-folds of index 2
reviewed by: Christopher Hacon
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s00229-007-0082-6
OpenAlex ID: W2067609392
cites work: Q4788838
cites work: Graded rings and special K3 surfaces
cites work: Computing certain Fano 3-folds
cites work: Fano 3-folds in codimension 4, Tom and Jerry. Part I
cites work: The Magma algebra system. I: The user language
cites work: Q4507515
cites work: Q4220921
cites work: On the plurigenera of minimal algebraic 3-folds with \(K\equiv 0\)
cites work: Q4026918
cites work: On 3-dimensional terminal singularities
cites work: Q3772328
cites work: Q4663330
cites work: On Fano indices of \(\mathbb Q\)-Fano 3-folds
DOI: 10.1007/S00229-007-0082-6
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description: scientific article; zbMATH DE number 5162386