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Tables of Contents for Algebraic Geometry
Chapter/Section Title
Page #
Page Count
Preface
vii
Acknowledgments
ix
Using This Book
xi
Part I: Examples of Varieties and Maps
Affine and Projective Varieties
3
14
A Note About Our Field
3
1
Affine Space and Affine Varieties
3
1
Projective Space and Projective Varieties
3
13
Linear Spaces
5
1
Finite Sets
6
2
Hypersurfaces
8
1
Analytic Subvarieties and Submanifolds
8
1
The Twisted Cubic
9
1
Rational Normal Curves
10
1
Determinantal Representation of the Rational Normal Curve
11
1
Another Parametrization of the Rational Normal Curve
11
1
The Family of Plane Conics
12
1
A Synthetic Construction of the Rational Normal Curve
13
1
Other Rational Curves
14
2
Varieties Defined over Subfields of K
16
1
A Note on Dimension, Smoothness, and Degree
16
1
Regular Functions and Maps
17
15
The Zariski Topology
17
1
Regular Functions on an Affine Variety
18
2
Projective Varieties
20
1
Regular Maps
21
11
The Veronese Map
23
1
Determinantal Representation of Veronese Varieties
24
1
Subvarieties of Veronese Varieties
24
1
The Segre Maps
25
2
Subvarieties of Segre Varieties
27
1
Products of Varieties
28
1
Graphs
29
1
Fiber Products
30
1
Combinations of Veronese and Segre Maps
30
2
Cones, Projections, and More About Products
32
9
Cones
32
1
Quadrics
33
1
Projections
34
3
More Cones
37
1
More Projections
38
1
Constructible Sets
39
2
Families and Parameter Spaces
41
7
Families of Varieties
41
1
The Universal Hyperplane
42
1
The Universal Hyperplane Section
43
1
Parameter Spaces of Hypersurfaces
44
1
Universal Families of Hypersurfaces
45
2
A Family of Lines
47
1
Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz
48
15
Generating Ideals
48
2
Ideals of Projective Varieties
50
1
Irreducible Varieties and Irreducible Decomposition
51
2
General Objects
53
4
General Projections
54
1
General Twisted Cubics
55
1
Double Point Loci
56
1
A Little Algebra
57
3
Restatements and Corollaries
60
3
Grassmannians and Related Varieties
63
9
Grassmannians
63
3
Subvarieties of Grassmannians
66
6
The Grassmannian G(1, 3)
67
1
An Analog of the Veronese Map
68
1
Incidence Correspondences
68
1
Varieties of Incident Planes
69
1
The Join of Two Varieties
70
1
Fano Varieties
70
2
Rational Functions and Rational Maps
72
16
Rational Functions
72
1
Rational Maps
73
2
Graphs of Rational Maps
75
2
Birational Isomorphism
77
2
The Quadric Surface
78
1
Hypersurfaces
79
1
Degree of a Rational Map
79
1
Blow-Ups
80
7
Blowing Up Points
81
1
Blowing Up Subvarieties
82
2
The Quadric Surface Again
84
1
The Cubic Scroll in P4
85
2
Unirationality
87
1
More Examples
88
10
The Join of Two Varieties
88
1
The Secant Plane Map
89
1
Secant Varieties
90
1
Trisecant Lines, etc.
90
1
Joins of Corresponding Points
91
1
Rational Normal Scrolls
92
1
Higher-Dimensional Scrolls
93
1
More Incidence Correspondences
94
1
Flag Manifolds
95
1
More Joins and Intersections
95
1
Quadrics of Rank 4
96
1
Rational Normal Scrolls II
97
1
Determinantal Varieties
98
16
Generic Determinantal Varieties
98
1
Segre Varieties
98
1
Secant Varieties of Segre Varieties
99
1
Linear Determinantal Varieties in General
99
12
Rational Normal Curves
100
3
Secant Varieties to Rational Normal Curves
103
2
Rational Normal Scrolls III
105
4
Rational Normal Scrolls IV
109
2
More General Determinantal Varieties
111
1
Symmetric and Skew-Symmetric Determinantal Varieties
112
2
Fano Varieties of Determinantal Varieties
112
2
Algebraic Groups
114
19
The General Linear Group GLnK
114
1
The Orthogonal Group SOnK
115
1
The Symplectic Group Sp2nK
116
1
Group Actions
116
7
PGLn+1K acts on Pn
116
1
PGL2K Acts on P2
117
1
PGL2K Acts on P3
118
1
PGL2K Acts on Pn
119
1
PGL3K Acts on P5
120
1
PGL3K Acts on P9
121
1
POnK Acts on Pn-1 (automorphisms of the Grassmannian)
122
1
PGLnK Acts on P(kKn)
122
1
Quotients
123
1
Quotients of Affine Varieties by Finite Groups
124
2
Quotients of Affine Space
125
1
Symmetric Products
126
1
Quotients of Projective Varieties by Finite Groups
126
7
Weighted Projective Spaces
127
6
Part II: Attributes of Varieties
Definitions of Dimension and Elementary Examples
133
18
Hypersurfaces
136
1
Complete Intersections
136
2
Immediate Examples
138
13
The Universal k-Plane
142
1
Varieties of Incident Planes
142
1
Secant Varieties
143
3
Secant Varieties in General
146
2
Joins of Varieties
148
1
Flag Manifolds
148
1
(Some) Schubert Varieties
149
2
More Dimension Computations
151
12
Determinantal Varieties
151
1
Fano Varieties
152
3
Parameter Spaces of Twisted Cubics
155
6
Twisted Cubics
155
1
Twisted Cubics on a General Surface
156
1
Complete Intersections
157
1
Curves of Type (a, b) on a Quadric
158
1
Determinantal Varieties
159
2
Group Actions
161
2
GL(V) Acts on SymdV and kV
161
1
PGLn+1K Acts on (Pn)1 and G(k, n)1
161
2
Hilbert Polynomials
163
11
Hilbert Functions and Polynomials
163
5
Hilbert Function of the Rational Normal Curve
166
1
Hilbert Function of the Veronese Variety
166
1
Hilbert Polynomials of Curves
166
2
Syzygies
168
6
Three Points in P2
170
1
Four Points in P2
171
1
Complete Intersections: Koszul Complexes
172
2
Smoothness and Tangent Spaces
174
12
The Zariski Tangent Space to a Variety
174
3
A Local Criterion for Isomorphism
177
4
Projective Tangent Spaces
181
3
Determinantal Varieties
184
2
Gauss Maps, Tangential and Dual Varieties
186
14
A Note About Characteristic
186
14
Gauss Maps
188
1
Tangential Varieties
189
1
The Variety of Tangent Lines
190
3
Joins of Intersecting Varieties
193
2
The Locus of Bitangent Lines
195
1
Dual Varieties
196
4
Tangent Spaces to Grassmannians
200
11
Tangent Spaces to Grassmannians
200
2
Tangent Spaces to Incidence Correspondences
202
1
Varieties of Incident Planes
203
1
The Variety of Secant Lines
204
1
Varieties Swept out by Linear Spaces
204
2
The Resolution of the Generic Determinantal Variety
206
2
Tangent Spaces to Dual Varieties
208
1
Tangent Spaces to Fano Varieties
209
2
Further Topics Involving Smoothness and Tangent Spaces
211
13
Gauss Maps on Curves
211
2
Osculating Planes and Associated Maps
213
1
The Second Fundamental Form
214
1
The Locus of Tangent Lines to a Variety
215
1
Bertini's Theorem
216
3
Blow-ups, Nash Blow-ups, and the Resolution of Singularities
219
3
Subadditivity of Codimensions of Intersections
222
2
Degree
224
15
Bezout's Theorem
227
4
The Rational Normal Curves
229
2
More Examples of Degrees
231
8
Veronese Varieties
231
2
Segre Varieties
233
1
Degrees of Cones and Projections
234
1
Joins of Varieties
235
2
Unirationality of Cubic Hypersurfaces
237
2
Further Examples and Applications of Degree
239
12
Multidegree of a Subvariety of a Product
239
1
Projective Degree of a Map
240
1
Joins of Corresponding Points
241
1
Varieties of Minimal Degree
242
1
Degrees of Determinantal Varieties
243
1
Degrees of Varieties Swept out by Linear Spaces
244
1
Degrees of Some Grassmannians
245
2
Harnack's Theorem
247
4
Singular Points and Tangent Cones
251
15
Tangent Cones
251
7
Tangent Cones to Determinantal Varieties
256
2
Multiplicity
258
2
Examples of Singularities
260
4
Resolution of Singularities for Curves
264
2
Parameter Spaces and Moduli Spaces
266
16
Parameter Spaces
266
2
Chow Varieties
268
5
Hilbert Varieties
273
2
Curves of Degree 2
275
3
Moduli Spaces
278
4
Plane Cubics
279
3
Quadrics
282
26
Generalities about Quadrics
282
1
Tangent Spaces to Quadrics
283
1
Plane Conics
284
1
Quadric Surfaces
285
2
Quadrics in Pn
287
2
Linear Spaces on Quadrics
289
6
Lines on Quadrics
290
1
Planes on Four-Dimensional Quadrics
291
2
Fano Varieties of Quadrics in General
293
2
Families of Quadrics
295
6
The Variety of Quadrics in P1
295
1
The Variety of Quadrics in P2
296
1
Complete Conics
297
2
Quadrics in Pn
299
2
Pencils of Quadrics
301
7
Hints for Selected Exercises
308
6
References
314
3
Index
317