Varieties of small degree with respect to codimension and ample vector bundles
type: Item
instance of: scholarly article
title: Varieties of small degree with respect to codimension and ample vector bundles
zbMATH Open document ID: 1170.14030
author: Carla Novelli
author: Antonio Lanteri
published in: Journal of the Mathematical Society of Japan
publication date: 2008-07-01T00:00:00Z
review text: Let \(X\) be a smooth complex projective variety of dimension \(n\), and \(\mathcal E\) an ample vector bundle of rank \(r \geq 2\) with a section whose zero locus is a smooth subvariety of \(X\) of the expected dimension. As in the classical case of ample divisors, often, if \(Z\) is special in some sense also the ambient manifold \(X\) has to be special, and in many cases a classification is possible. The paper under review deals with this setting assuming that \(\dim Z \geq 3\) and that an ample line bundle \(H\) on \(X\) restricts to a very ample line bundle \(H_Z\) on \(Z\) such that the degree \(d:=d(Z,H_Z)\) is small with respect to the codimension \(c\) of \(Z\) in \(\mathbb P(|H_Z|)\). More precisely the assumption is that \(d \leq 2c +1\), a bound which corresponds to the classification given by \textit{P. Ionescu} [Math. Ann. 271, 339--348 (1985; Zbl 0541.14032)]. The main result of the paper is a complete an effective classification of the triplets \((X, \mathcal E, H)\) as above; the main ingredients are results from adjunction theory and a careful study of the possible coexistence of different adjunction theoretic structures on the same manifold. The assumption on the degree can be rephrased in terms of the Fujita \(\Delta\)- genus of the pair \((Z,H_Z)\) as \(\Delta(Z,H_Z) \leq c\), and, from this point of view, the results in this paper can be considered as a continuation and a major improvement of results obtained by the authors in [Adv. Geom. 8, No. 2, 227--256 (2008; Zbl 1170.14031)].
reviewed by: Gianluca Occhetta
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14N30
zbMATH DE Number: 5294948
zbMATH Keywords: ample vector bundles
zbMATH Keywords: special varieties
zbMATH Keywords: adjunction theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06020341
OpenAlex ID: W2022348425
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DOI: 10.2969/JMSJ/06020341
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description: scientific article; zbMATH DE number 5294948