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Algebraic Geometry : A First Course - Joe Harris

Algebraic Geometry

A First Course

Hardcover Published: 1st December 1995
ISBN: 9780387977164
Number Of Pages: 330

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This book is intended to introduce students to algebraic geometry; to give them a sense of the basic objects considered, the questions asked about them, and the sort of answers one can expect to obtain. It thus emplasizes the classical roots of the subject. For readers interested in simply seeing what the subject is about, this avoids the more technical details better treated with the most recent methods. For readers interested in pursuing the subject further, this book will provide a basis for understanding the developments of the last half century, which have put the subject on a radically new footing. Based on lectures given at Brown and Harvard Universities, this book retains the informal style of the lectures and stresses examples throughout; the theory is developed as needed. The first part is concerned with introducing basic varieties and constructions; it describes, for example, affine and projective varieties, regular and rational maps, and particular classes of varieties such as determinantal varieties and algebraic groups. The second part discusses attributes of varieties, including dimension, smoothness, tangent spaces and cones, degree, and parameter and moduli spaces.

J. Harris

Algebraic Geometry

A First Course

"This book succeeds brilliantly by concentrating on a number of core topics (the rational normal curve, Veronese and Segre maps, quadrics, projections, Grassmannians, scrolls, Fano varieties, etc.) and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship."-MATHEMATICAL REVIEWS

Prefacep. vii
Acknowledgmentsp. ix
Using This Bookp. xi
Examples of Varieties and Maps
Affine and Projective Varietiesp. 3
A Note About Our Fieldp. 3
Affine Space and Affine Varietiesp. 3
Projective Space and Projective Varietiesp. 3
Linear Spacesp. 5
Finite Setsp. 6
Hypersurfacesp. 8
Analytic Subvarieties and Submanifoldsp. 8
The Twisted Cubicp. 9
Rational Normal Curvesp. 10
Determinantal Representation of the Rational Normal Curvep. 11
Another Parametrization of the Rational Normal Curvep. 11
The Family of Plane Conicsp. 12
A Synthetic Construction of the Rational Normal Curvep. 13
Other Rational Curvesp. 14
Varieties Defined over Subfields of Kp. 16
A Note on Dimension, Smoothness, and Degreep. 16
Regular Functions and Mapsp. 17
The Zariski Topologyp. 17
Regular Functions on an Affine Varietyp. 18
Projective Varietiesp. 20
Regular Mapsp. 21
The Veronese Mapp. 23
Determinantal Representation of Veronese Varietiesp. 24
Subvarieties of Veronese Varietiesp. 24
The Segre Mapsp. 25
Subvarieties of Segre Varietiesp. 27
Products of Varietiesp. 28
Graphsp. 29
Fiber Productsp. 30
Combinations of Veronese and Segre Mapsp. 30
Cones, Projections, and More About Productsp. 32
Conesp. 32
Quadricsp. 33
Projectionsp. 34
More Conesp. 37
More Projectionsp. 38
Constructible Setsp. 39
Families and Parameter Spacesp. 41
Families of Varietiesp. 41
The Universal Hyperplanep. 42
The Universal Hyperplane Sectionp. 43
Parameter Spaces of Hypersurfacesp. 44
Universal Families of Hypersurfacesp. 45
A Family of Linesp. 47
Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatzp. 48
Generating Idealsp. 48
Ideals of Projective Varietiesp. 50
Irreducible Varieties and Irreducible Decompositionp. 51
General Objectsp. 53
General Projectionsp. 54
General Twisted Cubicsp. 55
Double Point Locip. 56
A Little Algebrap. 57
Restatements and Corollariesp. 60
Grassmannians and Related Varietiesp. 63
Grassmanniansp. 63
Subvarieties of Grassmanniansp. 66
The Grassmannian G(1, 3)p. 67
An Analog of the Veronese Mapp. 68
Incidence Correspondencesp. 68
Varieties of Incident Planesp. 69
The Join of Two Varietiesp. 70
Fano Varietiesp. 70
Rational Functions and Rational Mapsp. 72
Rational Functionsp. 72
Rational Mapsp. 73
Graphs of Rational Mapsp. 75
Birational Isomorphismp. 77
The Quadric Surfacep. 78
Hypersurfacesp. 79
Degree of a Rational Mapp. 79
Blow-Upsp. 80
Blowing Up Pointsp. 81
Blowing Up Subvarietiesp. 82
The Quadric Surface Againp. 84
The Cubic Scroll in P[superscript 4]p. 85
Unirationalityp. 87
More Examplesp. 88
The Join of Two Varietiesp. 88
The Secant Plane Mapp. 89
Secant Varietiesp. 90
Trisecant Lines, etc.p. 90
Joins of Corresponding Pointsp. 91
Rational Normal Scrollsp. 92
Higher-Dimensional Scrollsp. 93
More Incidence Correspondencesp. 94
Flag Manifoldsp. 95
More Joins and Intersectionsp. 95
Quadrics of Rank 4p. 96
Rational Normal Scrolls IIp. 97
Determinantal Varietiesp. 98
Generic Determinantal Varietiesp. 98
Segre Varietiesp. 98
Secant Varieties of Segre Varietiesp. 99
Linear Determinantal Varieties in Generalp. 99
Rational Normal Curvesp. 100
Secant Varieties to Rational Normal Curvesp. 103
Rational Normal Scrolls IIIp. 105
Rational Normal Scrolls IVp. 109
More General Determinantal Varietiesp. 111
Symmetric and Skew-Symmetric Determinantal Varietiesp. 112
Fano Varieties of Determinantal Varietiesp. 112
Algebraic Groupsp. 114
The General Linear Group GL[subscript n]Kp. 114
The Orthogonal Group SO[subscript n]Kp. 115
The Symplectic Group Sp[subscript 2n]Kp. 116
Group Actionsp. 116
PGL[subscript n+1]K acts on P[superscript n]p. 116
PGL[subscript 2]K Acts on P[superscript 2]p. 117
PGL[subscript 2]K Acts on P[superscript 3]p. 118
PGL[subscript 2]K Acts on P[superscript n]p. 119
PGL[subscript 3]K Acts on P[superscript 5]p. 120
PGL[subscript 3]K Acts on P[superscript 9]p. 121
PO[subscript n]K Acts on P[superscript n-1] (automorphisms of the Grassmannian)p. 122
PGL[subscript n]K Acts on P([logical and superscript k]K[superscript n])p. 122
Quotientsp. 123
Quotients of Affine Varieties by Finite Groupsp. 124
Quotients of Affine Spacep. 125
Symmetric Productsp. 126
Quotients of Projective Varieties by Finite Groupsp. 126
Weighted Projective Spacesp. 127
Attributes of Varieties
Definitions of Dimension and Elementary Examplesp. 133
Hypersurfacesp. 136
Complete Intersectionsp. 136
Immediate Examplesp. 138
The Universal k-Planep. 142
Varieties of Incident Planesp. 142
Secant Varietiesp. 143
Secant Varieties in Generalp. 146
Joins of Varietiesp. 148
Flag Manifoldsp. 148
(Some) Schubert Varietiesp. 149
More Dimension Computationsp. 151
Determinantal Varietiesp. 151
Fano Varietiesp. 152
Parameter Spaces of Twisted Cubicsp. 155
Twisted Cubicsp. 155
Twisted Cubics on a General Surfacep. 156
Complete Intersectionsp. 157
Curves of Type (a, b) on a Quadricp. 158
Determinantal Varietiesp. 159
Group Actionsp. 161
GL (V) Acts on Sym[superscript d]V and [logical and superscript k]Vp. 161
PGL[subscript n+1]K Acts on (P[superscript n])[superscript l] and G(k, n)[superscript l]p. 161
Hilbert Polynomialsp. 163
Hilbert Functions and Polynomialsp. 163
Hilbert Function of the Rational Normal Curvep. 166
Hilbert Function of the Veronese Varietyp. 166
Hilbert Polynomials of Curvesp. 166
Syzygiesp. 168
Three Points in P[superscript 2]p. 170
Four Points in P[superscript 2]p. 171
Complete Intersections: Koszul Complexesp. 172
Smoothness and Tangent Spacesp. 174
The Zariski Tangent Space to a Varietyp. 174
A Local Criterion for Isomorphismp. 177
Projective Tangent Spacesp. 181
Determinantal Varietiesp. 184
Gauss Maps, Tangential and Dual Varietiesp. 186
A Note About Characteristicp. 186
Gauss Mapsp. 188
Tangential Varietiesp. 189
The Variety of Tangent Linesp. 190
Joins of Intersecting Varietiesp. 193
The Locus of Bitangent Linesp. 195
Dual Varietiesp. 196
Tangent Spaces to Grassmanniansp. 200
Tangent Spaces to Grassmanniansp. 200
Tangent Spaces to Incidence Correspondencesp. 202
Varieties of Incident Planesp. 203
The Variety of Secant Linesp. 204
Varieties Swept out by Linear Spacesp. 204
The Resolution of the Generic Determinantal Varietyp. 206
Tangent Spaces to Dual Varietiesp. 208
Tangent Spaces to Fano Varietiesp. 209
Further Topics Involving Smoothness and Tangent Spacesp. 211
Gauss Maps on Curvesp. 211
Osculating Planes and Associated Mapsp. 213
The Second Fundamental Formp. 214
The Locus of Tangent Lines to a Varietyp. 215
Bertini's Theoremp. 216
Blow-ups, Nash Blow-ups, and the Resolution of Singularitiesp. 219
Subadditivity of Codimensions of Intersectionsp. 222
Degreep. 224
Bezout's Theoremp. 227
The Rational Normal Curvesp. 229
More Examples of Degreesp. 231
Veronese Varietiesp. 231
Segre Varietiesp. 233
Degrees of Cones and Projectionsp. 234
Joins of Varietiesp. 235
Unirationality of Cubic Hypersurfacesp. 237
Further Examples and Applications of Degreep. 239
Multidegree of a Subvariety of a Productp. 239
Projective Degree of a Mapp. 240
Joins of Corresponding Pointsp. 241
Varieties of Minimal Degreep. 242
Degrees of Determinantal Varietiesp. 243
Degrees of Varieties Swept out by Linear Spacesp. 244
Degrees of Some Grassmanniansp. 245
Harnack's Theoremp. 247
Singular Points and Tangent Conesp. 251
Tangent Conesp. 251
Tangent Cones to Determinantal Varietiesp. 256
Multiplicityp. 258
Examples of Singularitiesp. 260
Resolution of Singularities for Curvesp. 264
Parameter Spaces and Moduli Spacesp. 266
Parameter Spacesp. 266
Chow Varietiesp. 268
Hilbert Varietiesp. 273
Curves of Degree 2p. 275
Moduli Spacesp. 278
Plane Cubicsp. 279
Quadricsp. 282
Generalities about Quadricsp. 282
Tangent Spaces to Quadricsp. 283
Plane Conicsp. 284
Quadric Surfacesp. 285
Quadrics in P[superscript n]p. 287
Linear Spaces on Quadricsp. 289
Lines on Quadricsp. 290
Planes on Four-Dimensional Quadricsp. 291
Fano Varieties of Quadrics in Generalp. 293
Families of Quadricsp. 295
The Variety of Quadrics in P[superscript 1]p. 295
The Variety of Quadrics in P[superscript 2]p. 296
Complete Conicsp. 297
Quadrics in P[superscript n]p. 299
Pencils of Quadricsp. 301
Hints for Selected Exercisesp. 308
Referencesp. 314
Indexp. 317
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780387977164
ISBN-10: 0387977163
Series: Graduate Texts in Mathematics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 330
Published: 1st December 1995
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 2.54
Weight (kg): 0.64

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