Finite geometries

347 Pages·2020·1.805 MB·English

by  Kiss G., Szonyi T

Finite Geometries

Finite Geometries
Gy(cid:127)orgy Kiss
Tam(cid:19)as Szo}nyi
CRCPress
Taylor&FrancisGroup
6000BrokenSoundParkwayNW,Suite300
BocaRaton,FL33487-2742
(cid:13)c 2020byTaylor&FrancisGroup,LLC
CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness
NoclaimtooriginalU.S.Governmentworks
Printedonacid-freepaper
InternationalStandardBookNumber-13:978-1-4987-2165-3(Hardback)
Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Rea-
sonable e(cid:11)orts have been made to publish reliable data and information, but the author
and publisher cannot assume responsibility for the validity of all materials or the conse-
quences of their use. The authors and publishers have attempted to trace the copyright
holdersofallmaterialreproducedinthispublicationandapologizetocopyrightholdersif
permissiontopublishinthisformhasnotbeenobtained.Ifanycopyrightmaterialhasnot
beenacknowledgedpleasewriteandletusknowsowemayrectifyinanyfuturereprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted,
reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other
means,nowknownorhereafterinvented,includingphotocopying,micro(cid:12)lming,andrecord-
ing,orinanyinformationstorageorretrievalsystem,withoutwrittenpermissionfromthe
publishers.
For permission to photocopy or use material electronically from this work, please access
www.copyright.com(http://www.copyright.com/)orcontacttheCopyrightClearanceCen-
ter, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-
for-pro(cid:12)t organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system
ofpaymenthasbeenarranged.
Trademark Notice:Productorcorporatenamesmaybetrademarksorregisteredtrade-
marks,andareusedonlyforidenti(cid:12)cationandexplanationwithoutintenttoinfringe.
Library of Congress Cataloging-in-Publication Data
Names:Kiss,Gy(cid:127)orgy(Mathematicsprofessor),author.jSz}onyi,T.,author.
Title:Finitegeometries/Gy(cid:127)orgyKissandTamasSz}onyi.
Description:BocaRaton:CRCPress,Taylor&FrancisGroup,2020.j
Includesbibliographicalreferencesandindex.
Identi(cid:12)ers:LCCN2019013231jISBN9781498721653
Subjects:LCSH:Finitegeometries.jCombinatorialgeometry.
Classi(cid:12)cation:LCCQA167.2.K572020jDDC516/.11--dc23
LCrecordavailableathttps://lccn.loc.gov/2019013231
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface vii
1 De(cid:12)nition of projective planes, examples 1
2 Basic properties of collineations and the Theorem of Baer 29
3 Coordinatization of projective planes 47
4 Projective spaces of higher dimensions 75
5 Higher dimensional representations 117
6 Arcs, ovals and blocking sets 133
7 (k;n)-arcs and multiple blocking sets 161
8 Algebraic curves and (cid:12)nite geometry 177
9 Arcs, caps, unitals and blocking sets in higher
dimensional spaces 199
10 Generalized polygons, M(cid:127)obius planes 229
11 Hyperovals 255
12 Some applications of (cid:12)nite geometry in combinatorics 275
13 Some applications of (cid:12)nite geometry in coding theory
and cryptography 301
Bibliography 321
Index 333
v

Preface
This is an extended and updated version of our earlier textbook V(cid:19)eges Ge-
ometri(cid:19)akpublishedin2001inHungarian.Theabovebookwasusedforvarious
courses on (cid:12)nite geometry at E(cid:127)otv(cid:127)os Lor(cid:19)and University, Budapest, and the
University of Szeged. We had di(cid:11)erent types of courses: for mathematics stu-
dents we had an overview type course touching the most important results in
all the chapters of the book without going into detail, for students interested
in (cid:12)nite geometries we had special courses going into more detail for example
on arcs, blocking sets, (k;n)-arcs and also higher dimensional analogues. For
students specialized in mathematics education we focused on con(cid:12)guration,
coordinatization and collineations, because that was related to their courses
on classical geometry. Finally, we have a (cid:12)nite geometry seminar (following
the tradition of Professor Ferenc K(cid:19)arteszi), where even more special and re-
centresultsarediscussed.Since2010thebookhasbeenusedattheUniversity
of Primorska, Koper, for the course Selected Topics in Finite Geometry.
Although this book is introductory, it also contains some more recent ma-
terial. In particular, we try to illustrate some methods in more detail, such
as the use of polynomials. We also try to indicate recent developments in the
topics covered by the book without proofs. Therefore, the book can be used
as a textbook for higher level undergraduate and lower level graduate courses
by choosing some of the topics appropriately, and also as a reference material
in the sense that it contains pointers to recently published works. We tried
to limit the background assumed in our book. However, linear algebra (over
(cid:12)nite (cid:12)elds), some knowledge of ((cid:12)nite) (cid:12)elds, polynomials, elementary group
theory and basic combinatorics is used in many places. Hungarian students
typically know some (classical) geometry, so we did not use much and tried
to brie(cid:13)y refer to the most important facts in the (cid:12)rst chapter.
Thereareimportantbooksonthetopic.TheauthorsgrewuponK(cid:19)arteszi’s
book Introduction to Finite Geometries [105] and several chapters follow it.
ThechapteroncoordinatizationandcollineationsfollowsthebookbyHughes
and Piper [95] and the appendix by Lombardo-Radice to Segre’s book Lec-
tures on Modern Geometry [151]. In many cases the books Projective Geome-
tries over Finite Fields, [88] Projective Spaces of Three Dimensions [87] by
Hirschfeld and the book General Galois Geometries [93] by Hirschfeld and
Thas,areveryusefulandcanbeusedtocheckthematerialonlymentionedin
ourtextbook.IntheninetieswehadaTempuscooperationwithmanyuniver-
sities and a lot of teaching material on topics related to the topics treated in
the present book were produced in those years. For example we had lectures
vii
viii Preface
oncryptographyandcodingtheory,whichiswhyweusedthebookProjective
Geometries [21] by Beutelspacher and Rosenbaum in the last chapter.
There are very important recent results by Ball on arcs in higher dimen-
sions and the MDS conjecture; they can be found in his book [8]. That book
also contains recent results about the connection of graph theory and (cid:12)nite
geometry. The books Generalized Quadrangles [140] by Payne and Thas, and
Generalized Polygons [177]byVanMaldeghemarethemonographsrelatedto
the topic in Chapter 10 of our book.
We would like to thank our colleagues and (former) students who helped
us in reading the di(cid:11)erent chapters. Here we have to mention P(cid:19)eter Sziklai
and Andr(cid:19)as G(cid:19)acs in the case of the Hungarian edition and Zolt(cid:19)an Bl(cid:19)azsik,
D(cid:19)aniel Lenger and Tam(cid:19)as H(cid:19)eger in the case of the English translation.
We are indebted to Tam(cid:19)as H(cid:19)eger for drawing the (cid:12)gures. Throughout the
years we worked closely with Aart Blokhuis (Eindhoven), G(cid:19)abor Korchm(cid:19)aros
(Potenza), Leo Storme (Ghent). We have used the discussions with them at
many places in this book.
We gratefully acknowledge the support of the NKFIH Grant NN114614
andtheeditorialworkofMikl(cid:19)osB(cid:19)ona.Finally,wethankourpublisherforthe
technical help in publishing the book.
1
De(cid:12)nition of projective planes, examples
In this book some familiarity with classical geometry will be assumed. The
classical results will not be used explicitly, but will just provide some back-
groundmotivationforsomeoftheresults.Probablyeveryonehaslearntabout
Euclidean planes. The classical projective plane comes from the classical Eu-
clidean plane by introducing ideal (or in(cid:12)nite) elements. Associated to a par-
allel class of lines we have an ideal (or in(cid:12)nite) point, and the ideal line (or
lineatin(cid:12)nity)consistsofallthein(cid:12)nitepoints.Theadvantageofintroducing
the classical projective plane is that there is no di(cid:11)erence between ordinary
and ideal points; two lines always intersect. In classical geometry typical the-
oremsstatethatundersomeconditionscertainlinespassthroughapoint(for
example, if we take a triangle, then the angle bisectors pass through a point)
or certain points are on a line. In some cases, the classical theorems use met-
ric properties of the plane (distances and angles), in other cases the order of
the points on a line, but there are interesting results that just use incidences
of points and lines. A notable example for this is the celebrated Theorem of
Desargues.
Theorem 1.1. Let A A A and B B B be two triangles in such a position
1 2 3 1 2 3
thatthelinesA B passthroughapointO.ConsiderthepointsA A \B B =
i i i j i j
C , where fi;j;kg=f1;2;3g. Then the points C ;C ;C are on a line t.
k 1 2 3
Less formally, when the two triangles are in perspective from the point
O then they are also in perspective from the line t. More details on Desar-
gues’ theorem can be found in Coxeter’s book [48], where similar theorems,
for example the Theorem of Pappus, are also discussed. These theorems will
also occur in our book, mainly in the context of (cid:12)nite planes and spaces. In
Chapters 2 and 3 we shall see how particular cases of Desargues’ theorem are
related to properties of the coordinate structure of the projective plane. We
shall also call the con(cid:12)guration of the ten points (A’s, B’s and C’s and O)
and the ten lines (the lines A B ; the sides of the two triangles and the line t)
i i
a closed Desargues con(cid:12)guration.
Letusnowstartthemoresystematicstudyofprojectiveanda(cid:14)neplanes.
De(cid:12)nition 1.2. A triple (P;E;I), where P and E are two disjoint non-empty
sets and I(cid:26)P (cid:2)E is an incidence relation, is called an incidence geometry.
1