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General Galois Geometries

verfasst von: J.W.P Hirschfeld, J.A. Thas

Verlag: Springer London

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume).
This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces.
General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.

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Inhaltsverzeichnis

Frontmatter

1. Quadrics

Abstract

Quadrics were introduced in Chapter 5 of PGOFF2. The properties of quadrics on a line were developed in Chapter 6 and in a plane in Chapter 7. The properties of quadrics in three dimensions were developed in Chapters 15 and 16 of FPSOTD. Quadrics in five dimensions were also considered in Chapters 15, 17 and 20. First the essential definitions are recalled.

J. W. P. Hirschfeld, J. A. Thas

2. Hermitian varieties

Abstract

In \({\rm F}_{q}={\rm GF}_(q)\), q square, the map \(x\mapsto x^{\sqrt{q}}=\bar{x}\) is an involutory automorphism. For a matrix \( A = ({a_{ij}}) \), write \( \bar{A} = ({\bar{a}_{ij}}) \).

J. W. P. Hirschfeld, J. A. Thas

3. Grassmann varieties

Abstract

Let \(\Pi_{r}\) be an r-space in PG(n, K), \(n \geq 3, 1\leq r\leq n-2\), and let \(P(x^{(0)}), P(x^{(1)}),\ldots,P(x^{(r)})\). with \( {x^{(i)}} = ({{x^{(i)}}_0}, {{x^{(i)}}_1}, . . . , {x{^{(i)}}_n}), \, {\rm be} \, r + 1 \) linearly independent points of \( \Pi_r\)

J. W. P. Hirschfeld, J. A. Thas

4. Veronese and Segre varieties

Abstract

The Veronese variety of all quadrics of \(PG(n, K), n \geq 1\), is the variety

$$\mathcal{V} = \left\{{\bf P}(x_0^2,x_1^2,\ldots,x_n^2,x_0x_1,\ldots,x_0x_n,x_1x_2,\ldots,x_1x_n,\ldots,x_{n-1}x_n)| {\bf P}(X) \mathrm{is\; a\; point\; of\; PG(\it n,K)}\right\}$$

of \(PG( N, K)\) with \(N \;=\;n(n+3)/2\), where \(X\;=\;(x_0,x_1,\ldots,x_n)\); then V is a variety of dimension n.

J. W. P. Hirschfeld, J. A. Thas

5. Embedded geometries

Abstract

A polar space S of (finite) rank n or projective index \(n-1, n\geq3\), is a set P of elements called points together with distinguished subsets called subspaces with the following properties.

J. W. P. Hirschfeld, J. A. Thas

6. Arcs and caps

Abstract

A (k; r, s; n, q)-set K is defined to be a set of k points in PG(n, q) with at most r points in any s-space such that K is not contained in a proper subspace. This is a slight variation on the definition of Section 3.3 of PGOFF2, where the last condition is not present. The large question is to describe all such sets. Four questions particularly are of interest.

J. W. P. Hirschfeld, J. A. Thas

7. Ovoids, spreads and m-systems of finite classical polar spaces

Abstract

In this chapter, ovoids, spreads and m-systems of finite classical polar spaces are introduced. Also SPG-reguli, SPG-systems, BLT-sets and sets with the BLT-property are defined. The main results on these topics are given, all without proof.

J. W. P. Hirschfeld, J. A. Thas

Backmatter

Titel: General Galois Geometries
verfasst von: J.W.P Hirschfeld
J.A. Thas
Copyright-Jahr: 2016
Verlag: Springer London
Electronic ISBN: 978-1-4471-6790-7
Print ISBN: 978-1-4471-6788-4
DOI: https://doi.org/10.1007/978-1-4471-6790-7

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