Combinatorial Veronese structures, their geometry, and problems of embeddability
type: Item
instance of: scholarly article
title: Combinatorial Veronese structures, their geometry, and problems of embeddability
zbMATH Open document ID: 1145.51002
published in: Results in Mathematics
publication date: 2008-06-10T00:00:00Z
review text: Let \(X\) be an arbitrary set. The point set of a Veronese structure \(\mathbf{V}_k(X)\) over \(X\) consists of all \(k\)-multisets with elements from \(X\). The authors present a detailed description of the geometry of the smallest reasonable Veronese structure \(\mathbf{V}_3(3)\) (various interpretations and representations, configurational axioms, realizability in a projective space, self-duality). It is shown that none of the configurations \(\mathbf{V}_k(n)\) and their duals, except \(\mathbf{V}_3(3)\), is embeddable into a Desarguesian projective space. This yields a large class of Desarguesian partial linear spaces which cannot be embedded into a Desarguesian projective space. Quite surprisingly, it turns out that some of these configurations can be completed to a linear space.
Mathematics Subject Classification ID: 51E14
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51A20
zbMATH DE Number: 5285869
zbMATH Keywords: partial linear space
zbMATH Keywords: Veronese structure
zbMATH Keywords: embeddability
author: M. Prażmowska
author: K. Prażmowski
reviewed by: József Szilasi
MaRDI profile type: Publication
full work available at URL: s00025-007-0279-8
OpenAlex ID: W1976989283
DOI: 10.1007/S00025-007-0279-8
description: scientific article; zbMATH DE number 5285869