Algebraic Geometry

A First Course

By: Joe Harris

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Hardcover | 21 December 1995

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356 Pages

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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.

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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

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Non-Fiction Mathematics Geometry Algebraic Geometry

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Non-Fiction Mathematics Geometry Algebraic Geometry

Preface	p. vii
Acknowledgments	p. ix
Using This Book	p. xi
Examples of Varieties and Maps
Affine and Projective Varieties	p. 3
A Note About Our Field	p. 3
Affine Space and Affine Varieties	p. 3
Projective Space and Projective Varieties	p. 3
Linear Spaces	p. 5
Finite Sets	p. 6
Hypersurfaces	p. 8
Analytic Subvarieties and Submanifolds	p. 8
The Twisted Cubic	p. 9
Rational Normal Curves	p. 10
Determinantal Representation of the Rational Normal Curve	p. 11
Another Parametrization of the Rational Normal Curve	p. 11
The Family of Plane Conics	p. 12
A Synthetic Construction of the Rational Normal Curve	p. 13
Other Rational Curves	p. 14
Varieties Defined over Subfields of K	p. 16
A Note on Dimension, Smoothness, and Degree	p. 16
Regular Functions and Maps	p. 17
The Zariski Topology	p. 17
Regular Functions on an Affine Variety	p. 18
Projective Varieties	p. 20
Regular Maps	p. 21
The Veronese Map	p. 23
Determinantal Representation of Veronese Varieties	p. 24
Subvarieties of Veronese Varieties	p. 24
The Segre Maps	p. 25
Subvarieties of Segre Varieties	p. 27
Products of Varieties	p. 28
Graphs	p. 29
Fiber Products	p. 30
Combinations of Veronese and Segre Maps	p. 30
Cones, Projections, and More About Products	p. 32
Cones	p. 32
Quadrics	p. 33
Projections	p. 34
More Cones	p. 37
More Projections	p. 38
Constructible Sets	p. 39
Families and Parameter Spaces	p. 41
Families of Varieties	p. 41
The Universal Hyperplane	p. 42
The Universal Hyperplane Section	p. 43
Parameter Spaces of Hypersurfaces	p. 44
Universal Families of Hypersurfaces	p. 45
A Family of Lines	p. 47
Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz	p. 48
Generating Ideals	p. 48
Ideals of Projective Varieties	p. 50
Irreducible Varieties and Irreducible Decomposition	p. 51
General Objects	p. 53
General Projections	p. 54
General Twisted Cubics	p. 55
Double Point Loci	p. 56
A Little Algebra	p. 57
Restatements and Corollaries	p. 60
Grassmannians and Related Varieties	p. 63
Grassmannians	p. 63
Subvarieties of Grassmannians	p. 66
The Grassmannian G(1, 3)	p. 67
An Analog of the Veronese Map	p. 68
Incidence Correspondences	p. 68
Varieties of Incident Planes	p. 69
The Join of Two Varieties	p. 70
Fano Varieties	p. 70
Rational Functions and Rational Maps	p. 72
Rational Functions	p. 72
Rational Maps	p. 73
Graphs of Rational Maps	p. 75
Birational Isomorphism	p. 77
The Quadric Surface	p. 78
Hypersurfaces	p. 79
Degree of a Rational Map	p. 79
Blow-Ups	p. 80
Blowing Up Points	p. 81
Blowing Up Subvarieties	p. 82
The Quadric Surface Again	p. 84
The Cubic Scroll in P[superscript 4]	p. 85
Unirationality	p. 87
More Examples	p. 88
The Join of Two Varieties	p. 88
The Secant Plane Map	p. 89
Secant Varieties	p. 90
Trisecant Lines, etc.	p. 90
Joins of Corresponding Points	p. 91
Rational Normal Scrolls	p. 92
Higher-Dimensional Scrolls	p. 93
More Incidence Correspondences	p. 94
Flag Manifolds	p. 95
More Joins and Intersections	p. 95
Quadrics of Rank 4	p. 96
Rational Normal Scrolls II	p. 97
Determinantal Varieties	p. 98
Generic Determinantal Varieties	p. 98
Segre Varieties	p. 98
Secant Varieties of Segre Varieties	p. 99
Linear Determinantal Varieties in General	p. 99
Rational Normal Curves	p. 100
Secant Varieties to Rational Normal Curves	p. 103
Rational Normal Scrolls III	p. 105
Rational Normal Scrolls IV	p. 109
More General Determinantal Varieties	p. 111
Symmetric and Skew-Symmetric Determinantal Varieties	p. 112
Fano Varieties of Determinantal Varieties	p. 112
Algebraic Groups	p. 114
The General Linear Group GL[subscript n]K	p. 114
The Orthogonal Group SO[subscript n]K	p. 115
The Symplectic Group Sp[subscript 2n]K	p. 116
Group Actions	p. 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
Quotients	p. 123
Quotients of Affine Varieties by Finite Groups	p. 124
Quotients of Affine Space	p. 125
Symmetric Products	p. 126
Quotients of Projective Varieties by Finite Groups	p. 126
Weighted Projective Spaces	p. 127
Attributes of Varieties
Definitions of Dimension and Elementary Examples	p. 133
Hypersurfaces	p. 136
Complete Intersections	p. 136
Immediate Examples	p. 138
The Universal k-Plane	p. 142
Varieties of Incident Planes	p. 142
Secant Varieties	p. 143
Secant Varieties in General	p. 146
Joins of Varieties	p. 148
Flag Manifolds	p. 148
(Some) Schubert Varieties	p. 149
More Dimension Computations	p. 151
Determinantal Varieties	p. 151
Fano Varieties	p. 152
Parameter Spaces of Twisted Cubics	p. 155
Twisted Cubics	p. 155
Twisted Cubics on a General Surface	p. 156
Complete Intersections	p. 157
Curves of Type (a, b) on a Quadric	p. 158
Determinantal Varieties	p. 159
Group Actions	p. 161
GL (V) Acts on Sym[superscript d]V and [logical and superscript k]V	p. 161
PGL[subscript n+1]K Acts on (P[superscript n])[superscript l] and G(k, n)[superscript l]	p. 161
Hilbert Polynomials	p. 163
Hilbert Functions and Polynomials	p. 163
Hilbert Function of the Rational Normal Curve	p. 166
Hilbert Function of the Veronese Variety	p. 166
Hilbert Polynomials of Curves	p. 166
Syzygies	p. 168
Three Points in P[superscript 2]	p. 170
Four Points in P[superscript 2]	p. 171
Complete Intersections: Koszul Complexes	p. 172
Smoothness and Tangent Spaces	p. 174
The Zariski Tangent Space to a Variety	p. 174
A Local Criterion for Isomorphism	p. 177
Projective Tangent Spaces	p. 181
Determinantal Varieties	p. 184
Gauss Maps, Tangential and Dual Varieties	p. 186
A Note About Characteristic	p. 186
Gauss Maps	p. 188
Tangential Varieties	p. 189
The Variety of Tangent Lines	p. 190
Joins of Intersecting Varieties	p. 193
The Locus of Bitangent Lines	p. 195
Dual Varieties	p. 196
Tangent Spaces to Grassmannians	p. 200
Tangent Spaces to Grassmannians	p. 200
Tangent Spaces to Incidence Correspondences	p. 202
Varieties of Incident Planes	p. 203
The Variety of Secant Lines	p. 204
Varieties Swept out by Linear Spaces	p. 204
The Resolution of the Generic Determinantal Variety	p. 206
Tangent Spaces to Dual Varieties	p. 208
Tangent Spaces to Fano Varieties	p. 209
Further Topics Involving Smoothness and Tangent Spaces	p. 211
Gauss Maps on Curves	p. 211
Osculating Planes and Associated Maps	p. 213
The Second Fundamental Form	p. 214
The Locus of Tangent Lines to a Variety	p. 215
Bertini's Theorem	p. 216
Blow-ups, Nash Blow-ups, and the Resolution of Singularities	p. 219
Subadditivity of Codimensions of Intersections	p. 222
Degree	p. 224
Bezout's Theorem	p. 227
The Rational Normal Curves	p. 229
More Examples of Degrees	p. 231
Veronese Varieties	p. 231
Segre Varieties	p. 233
Degrees of Cones and Projections	p. 234
Joins of Varieties	p. 235
Unirationality of Cubic Hypersurfaces	p. 237
Further Examples and Applications of Degree	p. 239
Multidegree of a Subvariety of a Product	p. 239
Projective Degree of a Map	p. 240
Joins of Corresponding Points	p. 241
Varieties of Minimal Degree	p. 242
Degrees of Determinantal Varieties	p. 243
Degrees of Varieties Swept out by Linear Spaces	p. 244
Degrees of Some Grassmannians	p. 245
Harnack's Theorem	p. 247
Singular Points and Tangent Cones	p. 251
Tangent Cones	p. 251
Tangent Cones to Determinantal Varieties	p. 256
Multiplicity	p. 258
Examples of Singularities	p. 260
Resolution of Singularities for Curves	p. 264
Parameter Spaces and Moduli Spaces	p. 266
Parameter Spaces	p. 266
Chow Varieties	p. 268
Hilbert Varieties	p. 273
Curves of Degree 2	p. 275
Moduli Spaces	p. 278
Plane Cubics	p. 279
Quadrics	p. 282
Generalities about Quadrics	p. 282
Tangent Spaces to Quadrics	p. 283
Plane Conics	p. 284
Quadric Surfaces	p. 285
Quadrics in P[superscript n]	p. 287
Linear Spaces on Quadrics	p. 289
Lines on Quadrics	p. 290
Planes on Four-Dimensional Quadrics	p. 291
Fano Varieties of Quadrics in General	p. 293
Families of Quadrics	p. 295
The Variety of Quadrics in P[superscript 1]	p. 295
The Variety of Quadrics in P[superscript 2]	p. 296
Complete Conics	p. 297
Quadrics in P[superscript n]	p. 299
Pencils of Quadrics	p. 301
Hints for Selected Exercises	p. 308
References	p. 314
Index	p. 317
Table of Contents provided by Syndetics. All Rights Reserved.

Algebraic Geometry

A First Course

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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