Table Of ContentUNITEXT 104
Elisabetta Fortuna · Roberto Frigerio
Rita Pardini
Projective
Geometry
Solved Problems and Theory Review
UNITEXT - La Matematica per il 3+2
Volume 104
Editor-in-chief
A. Quarteroni
Series editors
L. Ambrosio
P. Biscari
C. Ciliberto
M. Ledoux
W.J. Runggaldier
More information about this series at http://www.springer.com/series/5418
Elisabetta Fortuna Roberto Frigerio
(cid:129)
Rita Pardini
Projective Geometry
Solved Problems and Theory Review
123
Elisabetta Fortuna Rita Pardini
Dipartimento di Matematica Dipartimento di Matematica
Universitàdi Pisa Universitàdi Pisa
Pisa Pisa
Italy Italy
RobertoFrigerio
Dipartimento di Matematica
Universitàdi Pisa
Pisa
Italy
ISSN 2038-5722 ISSN 2038-5757 (electronic)
UNITEXT- La Matematica peril3+2
ISBN978-3-319-42823-9 ISBN978-3-319-42824-6 (eBook)
DOI 10.1007/978-3-319-42824-6
LibraryofCongressControlNumber:2016946954
TranslationfromtheItalianlanguageedition:Geometriaproiettiva.Problemirisoltierichiamiditeoria
by Elisabetta Fortuna, Roberto Frigerio and Rita Pardini, © Springer-Verlag Italia, Milano 2011.
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Preface
Projective geometry topics are taught in many degree programs in Mathematics,
Physics and Engineering, also because of their practical applications in areas such
as engineering, computer vision, architecture and cryptography. The literature on
this subject includes, besides the classical extensive treatises, by now very dated
from the point of view of language and terminology, some modern textbooks.
Among these we mention only ‘E. Casas-Alvero, Analytic Projective Geometry,
EMS Textbooks in Mathematics (2014)’, whose approach is close to ours and to
which we refer the reader for bibliographical references.
This is not one further textbook, in particular it has not been conceived for
sequential reading from the first to the last page; rather it aims to complement a
standard textbook, accompanying the reader in her/his journey through the subject
according to the philosophy of “learning by doing”. For this reason we make no
claimtobesystematic;ratherwehavetheambition,oratleastthehope,notonlyto
ease and reinforce the understanding of the material by presenting completely
worked out examples and applications of the theory, but also to awaken the
curiosity of the reader, to challenge her/him to find original solutions and develop
theabilityoflookingataquestionfromdifferentperspectives.Inaddition,thebook
presentsamongthesolvedproblemssomeclassicalgeometricresults,whoseproofs
areaccessiblewithintherelativelyelementarytechniquespresentedhere.Hopefully
these examples will encourage some readers to learn more about the topics treated
here and undertake the study of classical algebraic geometry.
IndeedthestartingpointoftheoriginalItalianversionofthisworkhasbeenour
experience in teaching the Projective Geometry course in the undergraduate
MathematicsdegreeprograminPisa,whichbroughtustorealizethedifficultiesthe
students encounter. However, the book contains also topics that, although usually
not treated in undergraduate courses, can be useful and interesting for a reader
wishing to learn more on the subject. Another feature of the text is that not only
complex hypersurfaces and algebraic curves are studied, as it is traditional, but
considerable attention is paid also to the real case.
v
vi Preface
The first chapter of the book contains a concise but exhaustive review of the
basicresultsofprojectivegeometry;theinterestedreadercanfindtheproofsinany
textbook on the subject. The goal of this first part is to provide the reader with an
overview ofthesubjectmatter and tofixthenotations andtheconceptsused later.
The following three chapters are collections of solved problems concerning,
respectively: the linear properties of projective spaces, the study of hypersurfaces
and plane algebraic curves and, finally, conics and quadrics. In solving the prob-
lemswehavegivenpreferenceneithertotheanalyticnortothesyntheticapproach,
but every time we have chosen the solution that seemed to us more interesting, or
moreelegantorquicker;sometimeswehavepresentedmorethanonesolution.The
difficulty level varies, ranging from merely computational exercises to more chal-
lenging theoretical problems. The exercises that in our opinion are harder are
markedwiththesymbol ,whosemeaningis:“takeiteasy,getyourselfacupof
coffeeortea,armyourselfwithpatienceanddetermination,andyouwillsucceedin
theend”.Inothercases,besidesgivingthesolutionthatisobjectivelythesimplest,
we have proposed alternative solutions that require longer or more involved
arguments, but have the merit of giving a deeper conceptual understanding or
highlighting connections with other phenomena that would not be evident other-
wise.Wehavemarkedthese,sotosay,“enlightening”solutionswiththesymbol .
Wehavetriedtoofferinthiswayaguidetoreadingthetext;indeed,webelievethat
thebestwayofusingit,oncethebasicnotionsgiveninthetheoryreviewhavebeen
acquired,istotrytosolvetheproblemspresentedonone'sown,usingthesolutions
that we give for checking and complementing one's personal work.
The necessary prerequisites are limited to some basic notions, mostly in linear
algebra,thatareusuallytaughtinthefirstyearcoursesoftheMathematics,Physics
or Engineering degree programs.
We are extremely grateful to Ciro Ciliberto for believing in this project,
encouraging us and constantly sharing with us his experience as mathematician,
book author and editor. We also wish to thank Dr. Francesca Bonadei of Springer
Italia for suggesting usto produce an English version of our book and assisting us
in its preparation.
Pisa, Italy Elisabetta Fortuna
April 2016 Roberto Frigerio
Rita Pardini
Contents
1 Theory Review.. .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Standard Notation.... ..... .... .... .... .... .... ..... .... 1
1.2 Projective Spaces and Subspaces, Projective Transformations .... 2
1.2.1 Projective Spaces and Subspaces... .... .... ..... .... 2
1.2.2 Projective Transformations ... .... .... .... ..... .... 3
1.2.3 Operations on Subspaces. .... .... .... .... ..... .... 3
1.2.4 The Projective Linear Group.. .... .... .... ..... .... 4
1.2.5 Fixed Points . ..... .... .... .... .... .... ..... .... 4
1.2.6 Degenerate Projective Transformations .. .... ..... .... 4
1.2.7 Projection Centred at a Subspace .. .... .... ..... .... 5
1.2.8 Perspectivities ..... .... .... .... .... .... ..... .... 5
1.3 Projective Frames and Homogeneous Coordinates .... ..... .... 5
1.3.1 General Position and Projective Frames . .... ..... .... 5
1.3.2 Systems of Homogeneous Coordinates .. .... ..... .... 6
1.3.3 Analytic Representation of a Projective
Transformation .... .... .... .... .... .... ..... .... 7
1.3.4 Change of Projective Frames.. .... .... .... ..... .... 7
1.3.5 Cartesian Representation of Subspaces .. .... ..... .... 7
1.3.6 Parametric Representation of Subspaces . .... ..... .... 8
1.3.7 Extension of Projective Frames.... .... .... ..... .... 9
1.3.8 Affine Charts. ..... .... .... .... .... .... ..... .... 9
1.3.9 Projective Closure of an Affine Subspace .... ..... .... 11
1.3.10 Projective Transformations and Change
of Coordinates..... .... .... .... .... .... ..... .... 11
1.4 Dual Projective Space and Duality.... .... .... .... ..... .... 12
1.4.1 Dual Projective Space... .... .... .... .... ..... .... 12
1.4.2 Duality Correspondence . .... .... .... .... ..... .... 13
1.4.3 Linear Systems of Hyperplanes.... .... .... ..... .... 13
1.4.4 Duality Principle... .... .... .... .... .... ..... .... 14
1.4.5 Dual Projectivity... .... .... .... .... .... ..... .... 15
vii
viii Contents
1.5 Projective Spaces of Dimension 1 .... .... .... .... ..... .... 15
1.5.1 Cross-Ratio.. ..... .... .... .... .... .... ..... .... 15
1.5.2 Symmetries of the Cross-Ratio .... .... .... ..... .... 16
1.5.3 Classification of the Projectivities of P1ðCÞ
and of P1ðRÞ. ..... .... .... .... .... .... ..... .... 17
1.5.4 Characteristic of a Projectivity. .... .... .... ..... .... 18
1.6 Conjugation and Complexification .... .... .... .... ..... .... 19
1.7 Affine and Projective Hypersurfaces... .... .... .... ..... .... 19
1.7.1 Homogeneous Polynomials... .... .... .... ..... .... 20
1.7.2 Affine and Projective Hypersurfaces .... .... ..... .... 21
1.7.3 Intersection of a Hypersurface with a Hyperplane... .... 23
1.7.4 Projective Closure of an Affine Hypersurface . ..... .... 24
1.7.5 Affine and Projective Equivalence of Hypersurfaces. .... 25
1.7.6 Intersection of a Hypersurface and a Line.... ..... .... 26
1.7.7 Tangent Space to a Hypersurface, Singular Points .. .... 27
1.7.8 Multiplicity of a Point of a Hypersurface .... ..... .... 28
1.7.9 Real Hypersurfaces . .... .... .... .... .... ..... .... 29
1.8 Quadrics... .... .... ..... .... .... .... .... .... ..... .... 31
1.8.1 First Notions and Projective Classification ... ..... .... 31
1.8.2 Polarity with Respect to a Quadric . .... .... ..... .... 33
1.8.3 Intersection of a Quadric with a Line ... .... ..... .... 35
1.8.4 Projective Quadrics in P2ðKÞ and in P3ðKÞ .. ..... .... 35
1.8.5 Quadrics in Rn .... .... .... .... .... .... ..... .... 39
1.8.6 Diametral Hyperplanes, Axes, Vertices.. .... ..... .... 43
1.8.7 Conics of R2. ..... .... .... .... .... .... ..... .... 45
1.8.8 Quadrics of R3 .... .... .... .... .... .... ..... .... 48
1.9 Plane Algebraic Curves .... .... .... .... .... .... ..... .... 51
1.9.1 Local Study of a Plane Algebraic Curve. .... ..... .... 51
1.9.2 The Resultant of Two Polynomials. .... .... ..... .... 53
1.9.3 Intersection of Two Curves... .... .... .... ..... .... 54
1.9.4 Inflection Points ... .... .... .... .... .... ..... .... 56
1.9.5 Linear Systems, Pencils.. .... .... .... .... ..... .... 56
1.9.6 Linear Conditions .. .... .... .... .... .... ..... .... 57
1.9.7 Pencils of Conics... .... .... .... .... .... ..... .... 58
2 Exercises on Projective Spaces.. .... .... .... .... .... ..... .... 61
3 Exercises on Curves and Hypersurfaces .. .... .... .... ..... .... 107
4 Exercises on Conics and Quadrics... .... .... .... .... ..... .... 173
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 263
About the Authors
ElisabettaFortuna wasborninPisain1955.In1977shereceivedherDiplomadi
LicenzainMathematicsfromScuolaNormaleSuperioreinPisa.Since2001sheis
Associate Professor at the University of Pisa. Her areas of research are real and
complex analytic geometry, real algebraic geometry, computational algebraic
geometry.
Roberto Frigerio was born in Como in 1977. In 2005 he received his Ph.D. in
Mathematics at Scuola Normale Superiore in Pisa. Since 2014 he is Associate
Professor at the University of Pisa. His primary scientific interests are focused on
low-dimensional topology, hyperbolic geometry and geometric group theory.
Rita Pardini was born in Lucca in 1960. She received her Ph.D. in Mathematics
from Scuola Normale Superiore in Pisa in 1990; she is Full Professor at the
UniversityofPisasince2004.Herareaofresearchisclassical algebraicgeometry,
in particular algebraic surfaces and their moduli, irregular varieties and coverings.
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