Arithmetically Cohen-Macaulay bundles and sets of points on general surfaces
type: Item
instance of: scholarly article
title: Arithmetically Cohen-Macaulay bundles and sets of points on general surfaces
zbMATH Open document ID: 1406.14033
author: Luca Chiantini
author: Rosa M. Miró-Roig
published in: Journal of Algebra
publication date: 2016-02-04T00:00:00Z
review text: Let \(S\subseteq\mathbb{P}^3\) be a general smooth surface of degree \(d\geq4\) with \(\text{Pic}(S)=\mathbb{Z}\). A vector bundle \(F\) on \(S\) is called \textit{initialized} if \(h^0(F)>h^0(F(-1))=0\). If \(F\) is a rank 2 initialized vector bundle, then it is stable if and only if \(c_1(F)>0\). A vector bundle \(F\) on \(S\) is said \textit{ACM} if it has no intermediate cohomology, that is if \(\mathrm{H}^1(S,F(t))=0\) for all \(t\in\mathbb{Z}\). The existence of rank 2 ACM bundles on \(S\) is related to the existence of arithmetically Gorenstein sets of points and also to the problem of pfaffian representation of forms. In the paper under review the authors give bounds for the first Chern class of rank \(2\) stable ACM bundles on \(S\). In particular they prove the following main results: -- if \(d\geq17\) and \(c\) is greater than a function of \(d\) growing like \(3\sqrt[3]{d}\), then a general surface of degree \(d\) contains no initialized, indecomposable ACM rank 2 vector bundles with first Chern class \(c\); -- for any \(2\leq c\leq 6\), there exists an initialized, indecomposable ACM rank 2 vector bundle with first Chern class \(c\) on a general surface of degree \(d\geq 2c\); -- for any integer \(c>0\), a general surface of degree \(d\gg c\) contains an initialized, indecomposable ACM rank 2 vector bundle with first Chern class \(c\).
reviewed by: Maria Chiara Brambilla
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14F17
zbMATH DE Number: 6539244
zbMATH Keywords: ACM bundles
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.12.005
OpenAlex ID: W2209230397
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DOI: 10.1016/J.JALGEBRA.2015.12.005
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description: scientific article; zbMATH DE number 6539244