Projective threefolds of small class
type: Item
instance of: scholarly article
title: Projective threefolds of small class
zbMATH Open document ID: 0615.14024
DOI: 10.1007/BF02941603
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1987-01-01T00:00:00Z
review text: Let \(X\subset {\mathbb{P}}^ n\) be a nondegenerate complex projective threefold of degree \(d\) and let \(\mu\) be its class, i.e. the number of hyperplanes tangent to X contained in a general pencil. By combining the class formula, expressing \(\mu\) in terms of the Betti numbers of X and of its linear sections, with recent results on surfaces and threefolds with low sectional genus, the authors determine the X as above with \(\mu \leq 8\). Moreover, restricting to a particular class of threefolds, which are \({\mathbb{P}}^ 1\)-bundles over a smooth surface, the authors classify those X for which \(\mu-d\) is small.
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 3995872
zbMATH Keywords: number of tangent hyperplanes
zbMATH Keywords: Betti numbers
zbMATH Keywords: linear sections
zbMATH Keywords: threefold
author: Antonio Lanteri
author: Cristina Turrini
MaRDI profile type: Publication
cites work: On the Homotopy Groups of Complex Projective Algebraic Manifolds.
cites work: Q3237568
cites work: Ancora sulla differenza tra la classe e l'ordine di una superficie algebrica
cites work: Q4195061
cites work: Algebraic geometry and local differential geometry
cites work: Q3329577
cites work: The topology of complex projective varieties after S. Lefschetz
cites work: On the class of a projective algebraic surface
cites work: Q4748259
cites work: Projective manifolds whose topology is strongly reflected in their hyperplane sections
cites work: Classification of Algebraic Surfaces with Sectional Genus less than or Equal to Six. II: Ruled Surfaces with dim ϕKx⊗L(x) = l
cites work: Classification of algebraic surfaces with sectional genus less than or equal to six. III: Ruled surfaces with $\dim \phi_{K_X \otimes L}(X)=2$.
cites work: Q3775665
cites work: Q3227238
cites work: Q3953893
cites work: Hyperplane sections of projective surfaces. I: The adjunction mapping
cites work: On the minimality of hyperplane sections of projective threefolds.
cites work: Q5623828
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description: scientific article; zbMATH DE number 3995872