3-folds in \({\mathbb{P}}^ 5\) of degree 12
type: Item
instance of: scholarly article
title: 3-folds in \({\mathbb{P}}^ 5\) of degree 12
zbMATH Open document ID: 0812.14026
DOI: 10.1007/BF02567709
author: Gerhard Edelmann
published in: Manuscripta Mathematica
publication date: 1995-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/155939
review text: The classification of (smooth) codimension-two subvarieties in \(\mathbb{P}^ 4\) and \(\mathbb{P}^ 5\) has gained interest since Ellingsrud and Peskine proved the finiteness of the number of components of the Hilbert scheme of non-general-type surfaces in \(\mathbb{P}^ 4\) (the analogous statement for \(\mathbb{P}^ 5\) has been proven by Braun, Ottaviani, Schneider and Schreyer). Two present research lines (too far from being convergent for the time being) are the classification of codimension-two subvarieties in \(\mathbb{P}^ 4\) and \(\mathbb{P}^ 5\) of small degree and, on the other hand, the search of an effective degree bound for non-general-type codimension-two subvarieties in \(\mathbb{P}^ 4\) and \(\mathbb{P}^ 5\). In the paper under review, the author gives one more step in the classification of smooth 3-folds in \(\mathbb{P}^ 5\), by treating the degree 12 case. More precisely, he gives a list of eight possible families and proves the actual existence of 3-folds for seven of the cases, the last case remaining undecided. In order to determine the invariants of the possible 3-folds, adjunction theory methods are used, and many possibilities need to be ruled out.
reviewed by: Enrique Arrondo
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14M07
zbMATH DE Number: 572235
zbMATH Keywords: codimension-two subvarieties
zbMATH Keywords: number of components of the Hilbert scheme of non-general-type surfaces
zbMATH Keywords: classification of smooth 3-folds
MaRDI profile type: Publication
cites work: Projective n-Folds of Log-General Type. I
cites work: On the adjunction theoretic classification of projective varieties
cites work: Threefolds of degree 9 and 10 in \({\mathbb{P}}^ 5\)
cites work: Q4027249
cites work: Some special properties of the adjunction theory for 3-folds in 𝑃⁵
cites work: On the adjunction theoretic classification of polarized varieties.
cites work: Q3134942
cites work: Classification of Buchsbaum subvarieties of codimension 2 in projective space.
cites work: Q3995221
cites work: Q3744295
cites work: Q3322234
cites work: 3-Mannigfaltigkeiten im \({\mathbb{P}}^ 5\) und ihre zugehörigen stabilen Garben
cites work: Über 2-codimensionale Untermannigfaltigkeiten vom Grad 7 in \({\mathbb{P}}^ 4\) und \({\mathbb{P}}^ 5\)
cites work: Liaison des variétés algébriques. I
description: scientific article; zbMATH DE number 572235