Remarks on higher-rank ACM bundles on hypersurfaces
type: Item
instance of: scholarly article
title: Remarks on higher-rank ACM bundles on hypersurfaces
zbMATH Open document ID: 1403.14078
author: G. V. Ravindra
author: Amit Tripathi
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2018-12-17T00:00:00Z
review text: This paper deals with the existence of arithmetically Cohen-Macaulay (aCM) bundles \(\mathcal{E}\) on smooth hypersurfaces \(X_d\subset \mathbb{P}^{n+1}\) of degree \(d\). It was conjectured by Buchweitz, Greuel and Schreyer [\textit{R. O. Buchweitz} et al., Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)] (motivated by the classification of aCM bundles on hyperquadrics) that if \(r:=\mathrm{rank}(\mathcal{E})<2^e\), where \(e:=\lfloor\frac{n-1}{2}\rfloor\), then \(\mathcal{E}\) splits as a direct sum of line bundles. Based on some splitting results, the first named author proposed a modified version of the aforementioned conjecture: for a general hypersurface \(X_d\subset \mathbb{P}^{n+1}\) of sufficiently high degree, an aCM bundle of rank \(r\) should split if \(r< 2^s\), where \(s:=\lfloor\frac{n+1}{2}\rfloor\). In this paper, the authors continue contributing to this conjecture proving that on a general hypersurface \(X_d\subset \mathbb{P}^5\) of degree \(d\geq 3\) there are no indecomposable rank-\(3\) aCM bundles with fewer than \(8\) generators (Theorem 1 and Corollary 1). Moving to the opposite case of aCM bundles with the maximum permitted number \(r.d\) of generators (i.e, the so-called Ulrich bundles), the authors prove that a smooth hypersurface of even degree does not support an Ulrich bundle of odd rank and determinant of the form \(\mathcal{O}_X(c)\) for \(c\in\mathbb{Z}\) (Theorem 2).
reviewed by: Joan Pons-Llopis
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 6994063
zbMATH Keywords: projective hypersurface
zbMATH Keywords: arithmetically Cohen-Macaulay bundle
zbMATH Keywords: Ulrich bundle
MaRDI profile type: Publication
full work available at URL: j.crma.2018.10.004
OpenAlex ID: W2898563019
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Wikidata QID: Q129031958
DOI: 10.1016/J.CRMA.2018.10.004
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description: scientific article; zbMATH DE number 6994063