\(n\)-dimensional Fano varieties of wild representation type
type: Item
instance of: scholarly article
title: \(n\)-dimensional Fano varieties of wild representation type
zbMATH Open document ID: 1311.14045
author: Joan Pons-Llopis
author: Rosa M. Miró-Roig
published in: Journal of Pure and Applied Algebra
publication date: 2014-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.3704
review text: A vector bundle \(F\) on a smooth projective variety \((X,\mathcal{O}_X)\) is called ACM if it has no intermediate cohomology, that is if \(\mathrm{H}^i(X,F\otimes\mathcal{O}_X(t))=0\) for all \(t\in\mathbb{Z}\) and \(0<i<\mathrm{dim}(X)\). Varieties supporting only a finite number of indecomposable ACM bundles, up to twist and isomorphism, are called of \textit{finite representation type} and are completely classified. On the opposite side, a variety is said of \textit{wild representation type} if there exist \(r\)-dimensional families of non-isomorphic indecomposable ACM sheaves, for arbitrarily large \(r\). In the paper under review the authors prove that all Fano blow-ups of \(\mathbb{P}^n\) at a finite number of points are of wild representation type. More precisely, for any \(r\geq n\), they construct families of simple ACM rank \(r\) vector bundles whose dimension is of order \(r^2\). In the case \(n=2\), i.e. when \(X\) is a Del Pezzo surface, these bundles are also proved to be \textit{B. Ulrich} [Math. Z. 188, 23--32 (1984; Zbl 0573.13013)], \(\mu\)-semistable and \(\mu\)-stable with respect to suitable polarizations.
reviewed by: Maria Chiara Brambilla
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 6290610
zbMATH Keywords: Fano varieties
zbMATH Keywords: Ulrich bundles
zbMATH Keywords: ACM vector bundles
zbMATH Keywords: representation type
MaRDI profile type: Publication
OpenAlex ID: W1533825181
arXiv ID: 1011.3704
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DOI: 10.1016/J.JPAA.2014.02.011
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description: scientific article