Algebraic geometry : a first course

Title:

Personal Author:

Harris, Joe

Series:

Graduate texts in mathematics ; v.133

Publication Information:

Berlin : Springer-Verlag, 1992

ISBN:

9780387977164

Subject Term:

Geometry, Algebraic

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000000091078	QA564.H36 1992	Open Access Book	Book	Searching... Unknown

This book is based on one-semester courses given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. It is intended to be, as the title suggests, a first introduction to the subject. Even so, a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century, the subject was practiced on a relatively concrete, down-to-earth level; the main objects of study were projective varieties, and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries. Ultimately, the subject was pushed beyond the limits of its foundations: by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners.

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Nowadays, geometry in its many forms suffers neglect at the undergraduate level. The curriculum is still dominated by abstract algebra, real analysis, and general topology, reflecting mathematical fashions of 40 years ago. In the early part of this century projective geometry was indeed a standard undergraduate subject. Then, between 1930 and 1970, for the sake of rigor and generality, algebraic geometry received a radical new foundation at the hands of Weil, Zariski, Serre, Grothendieck, and others. The concrete language of classical algebraic geometry was rendered obsolete, though its results, suitably reformulated, remain valid and important. Unfortunately, the language of modern algebraic geometry is much too abstract to present to undergraduates. Although Harris says the basis of his text is the many topics that are "nearly as well understood classically as they are in modern language," the book is really a bridge between the two perspectives. Some of the most important passages show the breaking points of classical theory. Modern theorems are stated, often without proof, when they address classical concerns. Detailed examinations of geometrical notions such as dimension and degree reveal the pathologies that justify more abstract formulations. The book is arranged as 22 informal, relatively independent, lectures. The almost zoological character of classical geometry is reflected in the wealth of examples filling nearly the first half of the book, including many of higher dimensions as well as the usual curves and surfaces. Cox, Little, and O'Shea offer an even more elementary introduction to algebraic geometry from an algebraic perspective. Their emphasis is on the algebra of polynomials that students meet first in high school. In college linear algebra courses students learn to manipulate linear equations in many varibles. This book treats the more difficult problem of manipulating higher-degree equations in many variables. Many procedures described here were discovered only in the last 30 years. These procedures form the basis for computer algebra systems such as Maple, Mathematica, and REDUCE; the reader is encouraged to use these systems to explore the various concepts. One chapter explores significant applications to robotics and automatic theorem proving. No background in abstract algebra is presumed. Because a student meets rings, ideals, and even groups here in an especially concrete setting, a course based on this text would be an attractive alternative to a traditional abstract algebra course, leaving the student well prepared for more advanced study of algebraic geometry. Both of these books are highly recommended for all academic libraries. D. V. Feldman; University of New Hampshire

Preface	p. vii
Acknowledgments	p. ix
Using This Book	p. xi
Part I Examples of Varieties and Maps
Lecture 1 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
Lecture 2 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
Lecture 3 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
Lecture 4 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
Lecture 5 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
Lecture 6 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
Lecture 7 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
Lecture 8 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
Lecture 9 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
Lecture 10 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
Part II Attributes of Varieties
Lecture 11 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
Lecture 12 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
Lecture 13 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
Lecture 14 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
Lecture 15 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
Lecture 16 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
Lecture 17 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
Lecture 18 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
Lecture 19 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
Lecture 20 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
Lecture 21 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
Lecture 22 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

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