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
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. |
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