On threefolds with the smallest nontrivial monodromy group
type: Item
instance of: scholarly article
title: On threefolds with the smallest nontrivial monodromy group
DOI: 10.2422/2036-2145.202204_004
author: Serge Lvovski
published in: Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V
publication date: 2024-09-22T00:00:00Z
review text: Using an adjunction-theoretic result due to \textit{A. J. Sommese} [J. Reine Angew. Math. 402, 211--220 (1989; Zbl 0675.14005)] together with a proposition from [Séminaire de géométrie algébrique du Bois-Marie 1967--1969. Groupes de monodromie en géométrie algébrique (SGA 7 II) par P. Deligne et N. Katz. Exposés X à XXII. Cham: Springer (1973; Zbl 0258.00005)], the author obtains a complete list of smooth threefolds for which the monodromy group, acting on the second cohomology group of its smooth hyperplane section, is \(\mathbb{Z}/2\mathbb{Z}\). The list includes the quadric \(Q\subset \mathbb{P}^4\), the Veronese variety or its isomorphic projection, the Segre variety and the blowup of \(\mathbb{P}^3\) at a point, embedded in \(\mathbb{P}^8\) by the complete linear system \(|2H-E|\), where \(H\) is the preimage of a plane in \(\mathbb{P}^3\) and \(E\) is the exceptional divisor, or an isomorphic projection of this variety. The possibility of such a classification was announced by \textit{F. L. Zak} [Lect. Notes Math. 1479, 273--280 (1991; Zbl 0793.14026)].
reviewed by: Vladimir P. Kostov
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14N30
zbMATH DE Number: 7917519
zbMATH Keywords: threefold
zbMATH Keywords: monodromy group
zbMATH Keywords: smooth hyperplane section
MaRDI profile type: Publication
cites work: The adjunction theory of complex projective varieties
cites work: Q5580319
cites work: Varieties with small dual varieties. II
cites work: Varieties with small dual varieties. I
cites work: On the structure of polarized manifolds with total deficiency one. I
cites work: On the structure of polarized manifolds with total deficiency one. II
cites work: Periods of integrals on algebraic manifolds. III: Some global differential-geometric properties of the period mapping
cites work: FANO 3-FOLDS. I
cites work: Q3050560
cites work: The topology of complex projective varieties after S. Lefschetz
cites work: On Curves and Surfaces with Projectively Equivalent Hyperplane Sections
cites work: Q3705555
cites work: Ample and spanned vector bundles with large \(c^ 2_ 1\) relative to \(c_ 2\) on surfaces
cites work: On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold.
cites work: An Enumeration of All Varieties of Degree 4
cites work: On the topology of certain algebraic threefold loci.
cites work: Q4012053
cites work: Q3140224
description: scientific article; zbMATH DE number 7917519