On the dimension of secant varieties
type: Item
instance of: scholarly article
title: On the dimension of secant varieties
zbMATH Open document ID: 1201.14038
DOI: 10.4171/JEMS/229
author: Luca Chiantini
author: Ciro Ciliberto
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2010-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0812.1904
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Consider an irreducible, projective non-degenerate variety \(X \subset \mathbb{P}^N\). Basic projective invariants related with \(X \subset \mathbb{P}^N\) are its \(k\)-secant varieties which are the Zariski closure of the linear spaces of dimension \(k\) in \(\mathbb{P}^N\) spanned by \(k+1\) points on \(X\). The study of the secant varieties of \(X \subset \mathbb{P}^N\) is relevant to understand the geometry, the topology and the equations defining \(X \subset \mathbb{P}^N\). As usual in projective geometry a notion of expected dimension of the \(k\)-secant variety is known and the problem of classification of \textit{defective} varieties (the ones not reaching the expected dimension) is of interest. In the introduction of the paper under review precise references on this topic are presented. A main ingredient is the so-called Zak's theorem on tangencies bounding the dimension of the subvariety along which a linear spaces is tangent to \(X \subset \mathbb{P}^N\). The paper proposes a generalization of this theorem (see Theorem 6.1) by introducing the notion of \(J_k\)-tangency which generalizes the one of \(J\)-tangency (now \(J_1\)-tangency) of Zak. The authors consider limits of linear spaces of dimension \(s \leq k\) instead of just limits of lines (go to Definition 5.1 for the precise definition). Under suitable regularity conditions they can use this notions of \(J_k\)-tangency and the theorem of tangencies to extend Zak's theorem on linear normality (see Theorem 7.5), to revisit the classification of Scorza varieties in these terms and to make some speculations on possible generalizations of Hartshorne's conjecture on complete intersections.
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 5779487
zbMATH Keywords: secant varieties
zbMATH Keywords: tangential projections
zbMATH Keywords: defective varieties
reviewed by: Roberto Muñoz
MaRDI profile type: Publication
arXiv ID: 0812.1904
cites work: Joins and higher secant varieties.
cites work: On the Homotopy Groups of Complex Projective Algebraic Manifolds.
cites work: Q4852361
cites work: Ranks of tensors, secant varieties of Segre varieties and fat points
cites work: Weakly defective varieties
cites work: Threefolds with degenerate secant variety: On a theorem of G. Scorza.
cites work: Q5487000
cites work: ON THE CONCEPT OF k-SECANT ORDER OF A VARIETY
cites work: Q4408040
cites work: Terracini's Lemma and the Secant Variety of a Curve
cites work: Varieties with Small Secant Varieties: The Extremal Case
cites work: A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings
cites work: Algebraic geometry of Bayesian networks
cites work: Varieties of small codimension in projective space
cites work: Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman
cites work: Conic-connected manifolds
cites work: Tangential projections and secant defective varieties
cites work: Varieties of sums of power
cites work: Varieties with quadratic entry locus. I
cites work: PROJECTIONS OF ALGEBRAIC VARIETIES
cites work: Q3140224
OpenAlex ID: W2080397230
description: scientific article