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E-raamat: Projective Geometry: Solved Problems and Theory Review

Elisabetta Fortuna, Roberto Frigerio, Rita Pardini

Formaat: PDF+DRM
Sari: La Matematica per il 3+2 104
Ilmumisaeg: 17-Dec-2016
Kirjastus: Springer International Publishing AG
Keel: eng
ISBN-13: 9783319428246

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Formaat: PDF+DRM
Sari: La Matematica per il 3+2 104
Ilmumisaeg: 17-Dec-2016
Kirjastus: Springer International Publishing AG
Keel: eng
ISBN-13: 9783319428246

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This book starts with a concise but rigorous overview of the basic notions of projective geometry, using straightforward and modern language. The goal is not only to establish the notation and terminology used, but also to offer the reader a quick survey of the subject matter. In the second part, the book presents more than 200 solved problems, for many of which several alternative solutions are provided. The level of difficulty of the exercises varies considerably: they range from computations to harder problems of a more theoretical nature, up to some actual complements of the theory. The structure of the text allows the reader to use the solutions of the exercises both to master the basic notions and techniques and to further their knowledge of the subject, thus learning some classical results not covered in the first part of the book. The book addresses the needs of undergraduate and graduate students in the theoretical and applied sciences, and will especially benefit those r

eaders with a solid grasp of elementary Linear Algebra.

1 Theory review.- 2 Exercises on projective spaces.- 3 Exercises on curves and hypersurfaces.- 4 Esercises on conics and quadrics.

Arvustused

This is a modern textbook on classic projective geometry over (mostly) real and complex vectorspaces. The book will also serve well as a source and reference for an undergraduate lecture plus exercise course on projective geometry. (Hans-Peter Schröcker, zbMATH 1361.51004, 2017)

1 Theory Review

(60)

1.1 Standard Notation

(1)

1.2 Projective Spaces and Subspaces, Projective Transformations

(3)

1.2.1 Projective Spaces and Subspaces

(1)

1.2.2 Projective Transformations

(1)

1.2.3 Operations on Subspaces

(1)

1.2.4 The Projective Linear Group

(1)

1.2.5 Fixed Points

(1)

1.2.6 Degenerate Projective Transformations

(1)

1.2.7 Projection Centred at a Subspace

(1)

1.2.8 Perspectivities

(1)

1.3 Projective Frames and Homogeneous Coordinates

(7)

1.3.1 General Position and Projective Frames

(1)

1.3.2 Systems of Homogeneous Coordinates

(1)

1.3.3 Analytic Representation of a Projective Transformation

(1)

1.3.4 Change of Projective Frames

(1)

1.3.5 Cartesian Representation of Subspaces

(1)

1.3.6 Parametric Representation of Subspaces

(1)

1.3.7 Extension of Projective Frames

(1)

1.3.8 Affine Charts

(2)

1.3.9 Projective Closure of an Affine Subspace

(1)

1.3.10 Projective Transformations and Change of Coordinates

(1)

1.4 Dual Projective Space and Duality

(3)

1.4.1 Dual Projective Space

(1)

1.4.2 Duality Correspondence

(1)

1.4.3 Linear Systems of Hyperplanes

(1)

1.4.4 Duality Principle

(1)

1.4.5 Dual Projectivity

(1)

1.5 Projective Spaces of Dimension 1

(4)

1.5.1 Cross-Ratio

(1)

1.5.2 Symmetries of the Cross-Ratio

(1)

1.5.3 Classification of the Projectivities of P'(C) and of P1(R)

(1)

1.5.4 Characteristic of a Projectivity

(1)

1.6 Conjugation and Complexification

(1)

1.7 Affine and Projective Hypersurfaces

(12)

1.7.1 Homogeneous Polynomials

(1)

1.7.2 Affine and Projective Hypersurfaces

(2)

1.7.3 Intersection of a Hypersurface with a Hyperplane

(1)

1.7.4 Projective Closure of an Affine Hypersurface

(1)

1.7.5 Affine and Projective Equivalence of Hypersurfaces

(1)

1.7.6 Intersection of a Hypersurface and a Line

(1)

1.7.7 Tangent Space to a Hypersurface, Singular Points

(1)

1.7.8 Multiplicity of a Point of a Hypersurface

(1)

1.7.9 Real Hypersurfaces

(2)

1.8 Quadrics

(20)

1.8.1 First Notions and Projective Classification

(2)

1.8.2 Polarity with Respect to a Quadric

(2)

1.8.3 Intersection of a Quadric with a Line

(1)

1.8.4 Projective Quadrics in P2(K) and in P3(K)

(4)

1.8.5 Quadrics in Rn

(4)

1.8.6 Diametral Hyperplanes, Axes, Vertices

(2)

1.8.7 Conies of R2

(3)

1.8.8 Quadrics of R3

(3)

1.9 Plane Algebraic Curves

(10)

1.9.1 Local Study of a Plane Algebraic Curve

(2)

1.9.2 The Resultant of Two Polynomials

(1)

1.9.3 Intersection of Two Curves

(2)

1.9.4 Inflection Points

(1)

1.9.5 Linear Systems, Pencils

(1)

1.9.6 Linear Conditions

(1)

1.9.7 Pencils of Conies

(3)

2 Exercises on Projective Spaces

(46)

3 Exercises on Curves and Hypersurfaces

107

(66)

4 Exercises on Conies and Quadrics

173

(90)

Index

263

Elisabetta Fortuna was born in Pisa in 1955. In 1977 she received her Diploma di Licenza in Mathematics from Scuola Normale Superiore in Pisa. Since 2001 she is 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 University of Pisa since 2004. Her area of research is classical algebraic geometry, in particular algebraic surfaces and their moduli, irregular varieties and coverings.

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E-raamat: Projective Geometry: Solved Problems and Theory Review

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