Pseudo-embeddings and quadratic sets of quadrics
type: Item
instance of: scholarly article
title: Pseudo-embeddings and quadratic sets of quadrics
zbMATH Open document ID: 1486.51006
author: Bart De Bruyn
author: Mou Gao
published in: Designs, Codes and Cryptography
publication date: 2022-01-19T00:00:00Z
review text: This paper gives a local characterisation of the intersection of two quadrics. More precisely, the authors consider quadratic sets of a non-singular quadric \(Q\), meaning a set of points of \(Q\) with the property that it intersects each subspace of \(Q\) in a possibly singular quadric. The standard example is then given by intersecting \(Q\) with another quadric in the same projective space. In the current paper, the quadrics that are considered are the Klein quadric \(Q^+(5,2)\), the parabolic quadric \(Q(6,2)\) and the elliptic quadric \(Q^-(7,2)\), and the result is that the quadratic sets are all given by intersecting with a quadric. It is also mentioned that this result is no longer true for \(Q^+(5,q)\) with \(q\geq 3\), showing that these small cases behave differently. The paper arose as a by-product of other investigations, related to the study of certain line sets of \(\mathsf{PG}(3,q)\), which gives points on the Klein quadric \(Q^+(5,q)\), yielding a new example. The method uses pseudo-embeddings of the geometry formed by the points of the quadric \(Q\) and its planes.
reviewed by: Anneleen De Schepper
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 7459700
zbMATH Keywords: quadric
zbMATH Keywords: quadratic set
zbMATH Keywords: pseudo-hyperplane
zbMATH Keywords: (universal) pseudo-embedding
zbMATH Keywords: pseudo-generating set
MaRDI profile type: Publication
full work available at URL: s10623-021-00971-8
OpenAlex ID: W3211460212
cites work: Q3227204
cites work: Q4376259
cites work: Ovoids of PG(3,q ), q Even, with a Conic Section
cites work: Ensembles quadratiques des espaces projectifs
cites work: On the foundations of polar geometry
cites work: Quadratic sets on the Klein quadric
cites work: Pseudo-embeddings and pseudo-hyperplanes
cites work: Pseudo-embeddings of the (point, \(k\)-spaces)-geometry of \(\mathrm{PG}(n,2)\) and projective embeddings of \(\mathrm{DW}(2n-1,2)\)
cites work: The homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle \(H(3,4)\)
cites work: Q3227205
cites work: A characterization of the family of secant lines to a hyperbolic quadric in \(\mathrm{PG} ( 3 , q )\), \(q\) odd
cites work: Linear Algebra and Geometry
cites work: On Veldkamp lines
cites work: Foundations of Incidence Geometry
DOI: 10.1007/S10623-021-00971-8
description: scientific article; zbMATH DE number 7459700