Zak's theorem on superadditivity
type: Item
instance of: scholarly article
title: Zak's theorem on superadditivity
zbMATH Open document ID: 0823.14039
DOI: 10.1007/BF02559525
author: Audun Holme
author: Joel L. Roberts
published in: Arkiv för Matematik
publication date: 1995-11-05T00:00:00Z
review text: The authors give a proof of Zak's theorem: Let \(X\) be a nonsingular, closed subvariety (reduced, irreducible) in \(\mathbb{P}^ N\). Assume that the \(p\)-secant variety \(S^ k (X)\) is not equal to \(\mathbb{P}^ N\). Then \[ \delta_{p+1} (X)\geq \delta_ p (X)+ \delta_ 1 (X), \] where the numbers \[ \delta_ k (X)= \dim S^{k-1} (X)+ \dim X+1- \dim S^ k (X) \] are the secant defects of \(X\) in \(\mathbb{P}^ N\). The proof differs from that of Zak in that it is of an infinitesimal nature. It grew out of an attempt to understand and clarify the proof given by Zak. An earlier such attempt was made by B. Fantechi who gave a proof of the same theorem following closer to the one indicated by Zak. - - The article also contains some results on a more general statement, often called Zak's conjecture.
reviewed by: Dan Laksov
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14M99
Mathematics Subject Classification ID: 51M35
zbMATH DE Number: 605119
zbMATH Keywords: secant variety
zbMATH Keywords: secant defects
zbMATH Keywords: Zak's conjecture
MaRDI profile type: Publication
cites work: Joins and higher secant varieties.
cites work: On the superadditivity of secant defects
cites work: Varieties with Small Secant Varieties: The Extremal Case
cites work: Linear systems of hyperplane sections on varieties of low codimension
cites work: Q3698937
description: scientific article; zbMATH DE number 605119