The representation type of rational normal scrolls
type: Item
instance of: scholarly article
title: The representation type of rational normal scrolls
zbMATH Open document ID: 1268.14014
publication date: 2013-04-02T00:00:00Z
review text: Let \((X,\mathcal{O}_X(1))\) be a polarized projective variety of dimension \(n\). A sheaf \(E\) on \(X\) is called Arithmetically Cohen-Macaulay (ACM), if all its middle cohomologies vanish. That is, \(H^i(X,E(k))=0\) for all \(k\) and \(0<i<n\), where as usual, one writes \(E(k)\) to mean \(E\otimes\mathcal{O}_X^{\otimes k}\). For such an \(E\), one can see easily that the number of generators of \(\bigoplus H^0(X,E(k))\) is at most \(\deg X\cdot \mathrm{rk}\, E\). An \(E\) where equality holds are called Ulrich bundles and they have been studied intensely and introduced by \textit{B. Ulrich} [Math. Z. 188, 23--32 (1984; Zbl 0573.13013)]. The paper under review shows that on most rational normal scrolls there are arbitrarily large rank and dimension families of indecomposable Ulrich bundles. The exceptions, a small number, were previously known to have not many indecomposable Ulrich bundles [\textit{R.-O. Buchweitz, G.-M. Greuel} and \textit{F.-O. Schreyer}, Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)].
reviewed by: N. Mohan Kumar
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M20
zbMATH DE Number: 6149393
zbMATH Keywords: scroll
zbMATH Keywords: representation type
zbMATH Keywords: Ulrich bundles
zbMATH Keywords: arithmetic Cohen-Macaulay sheaves
author: Rosa M. Miró-Roig
MaRDI profile type: Publication
full work available at URL: s12215-013-0113-y
OpenAlex ID: W2008301549
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DOI: 10.1007/S12215-013-0113-Y
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6149393