Verfügbarkeit: Algebraic geometry

Algebraic geometry: a first course

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Bibliographische Detailangaben
1. Verfasser:	Harris, Joe 1951- (VerfasserIn)
Format:	Buch
Sprache:	Undetermined
Veröffentlicht:	New York [u.a.] Springer [2003]
Ausgabe:	[reprint]
Schriftenreihe:	Graduate texts in mathematics 133
Schlagworte:	Algebraische Geometrie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XIX, 328 S. graph. Darst.
ISBN:	0387977163 9780387977164 3540977163

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Datensatz im Suchindex

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adam_text	Contents Preface vii Acknowledgments ix Using This Book xi Part i: Examples of Varieties and Maps Lecture 1 Affine and Projective Varieties 3 A Note About Our Field 3 Affine Space and Affine Varieties 3 Projective Space and Projective Varieties 3 Linear Spaces 5 Finite Sets 6 Hypersurfaces 8 Analytic Subvarieties and Submanifolds 8 The Twisted Cubic 9 Rational Normal Curves 10 Determinantal Representation of the Rational Normal Curve 11 Another Parametrization of the Rational Normal Curve 11 The Family of Plane Conies 12 A Synthetic Construction of the Rational Normal Curve 13 Other Rational Curves 14 Varieties Defined over Subfields of К 16 A Note on Dimension, Smoothness, and Degree 16 Lecture 2 Regular Functions and Maps 17 The Zariski Topology 17 Regular Functions on an Affine Variety 18 Contents Projective Varieties 20 Regular Maps 21 The Veronese Map 23 Determinantal Representation of Veronese Varieties 24 Subvarieties of Veronese Varieties 24 The Segre Maps 25 Subvarieties of Segre Varieties 27 Products of Varieties 28 Graphs 29 Fiber Products 30 Combinations of Veronese and Segre Maps 30 Lecture 3 Cones, Projections, and More About Products 32 Cones 32 Quadrics 33 Projections 34 More Cones 37 More Projections 38 Constructible Sets 39 Lecture 4 Families and Parameter Spaces 41 Families of Varieties 41 The Universal Hyperplane 42 The Universal Hyperplane Section 43 Parameter Spaces of Hypersurfaces 44 Universal Families of Hypersurfaces 45 A Family of Lines 47 Lecture 5 Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz 48 Generating Ideals 48 Ideals of Projective Varieties 50 Irreducible Varieties and Irreducible Decomposition 51 General Objects 53 General Projections 54 General Twisted Cubics 55 Double Point Loci 56 A Little Algebra 57 Restatements and Corollaries 60 Lecture 6 Grassmannians and Related Varieties 63 Grassmannians 63 Subvarieties of Grassmannians 66 Contents xv The Grassmannian G(l, 3) 67 An Analog of the Veronese Map 68 Incidence Correspondences 68 Varieties of Incident Planes 69 The Join of Two Varieties 70 Fano Varieties 70 Lecture 7 Rational Functions and Rational Maps 72 Rational Functions 72 Rational Maps 73 Graphs of Rational Maps 75 Birational Isomorphism 77 The Quadric Surface 78 Hypersurfaces 79 Degree of a Rational Map 79 BIow-Ups 80 Blowing Up Points 81 Blowing Up Subvarieties 82 The Quadric Surface Again 84 The Cubic Scroll in P4 85 Unirationality 87 Lecture 8 More Examples 88 The Join of Two Varieties 88 The Secant Plane Map 89 Secant Varieties 90 Trisecant Lines, etc. 90 Joins of Corresponding Points 91 Rational Normal Scrolls 92 Higher-Dimensional Scrolls 93 More Incidence Correspondences 94 Flag Manifolds 95 More Joins and Intersections 95 Quadrics of Rank 4 96 Rational Normal Scrolls II 97 Lecture 9 Determinantal Varieties 98 Generic Determinantal Varieties 98 Segre Varieties 98 Secant Varieties of Segre Varieties 99 Linear Determinantal Varieties in General 99 Rational Normal Curves 100 Secant Varieties to Rational Normal Curves 103 Rational Normal Scrolls III Ю5 Contents Rational Normal Scrolls IV 109 More General Determinantal Varieties 111 Symmetrie and Skew-Symmetric Determinantal Varieties 112 Fano Varieties of Determinantal Varieties 112 Lecture 10 Algebraic Groups 114 The General Linear Group GL„K 114 The Orthogonal Group SO„K 115 The Symplectic Group Sp2nX 116 Group Actions 116 PGL.+jKactsonP 116 PGL2X Acts on P2 117 PGL2K Acts on P3 118 PGL2K Acts on P 119 PGL3K Acts on P5 120 PGL3K Acts on P9 121 РО„К Acts on P ~ (automorphisms of the Grassmannian) 122 PGL„K Acts on P(AkKn) 122 Quotients 123 Quotients of Affine Varieties by Finite Groups 124 Quotients of Affine Space 125 Symmetric Products 126 Quotients of Projective Varieties by Finite Groups 126 Weighted Projective Spaces 127 Part ii: Attributes of Varieties Lecture 11 Definitions of Dimension and Elementary Examples 133 Hypersurfaces 136 Complete Intersections 136 Immediate Examples 138 The Universal fc-Plane 142 Varieties of Incident Planes 142 Secant Varieties 143 Secant Varieties in General 146 Joins of Varieties 148 Flag Manifolds 148 (Some) Schubert Varieties 149 Lecture 12 M ore Dimension Computations 151 Determinantal Varieties 151 Fano Varieties 152 Parameter Spaces of Twisted Cubics 155 Twisted Cubics 155 Contents Twisted Cubics on a General Surface 156 Complete Intersections 157 Curves of Type (a, b) on а Quadric 158 Determinantal Varieties 159 Group Actions 161 GLtKlActsonSynťVandA^K 161 PGL^, К Acts on (Ρ ) and G(k, «) 161 Lecture 13 Hubert Polynomials 163 Hubert Functions and Polynomials 163 Hubert Function of the Rational Normal Curve 166 Hubert Function of the Veronese Variety 166 Hubert Polynomials of Curves 166 Syzygies 168 Three Points in P2 170 Four Points in P2 171 Complete Intersections: Koszul Complexes 172 Lecture 14 Smoothness and Tangent Spaces 174 The Zariski Tangent Space to a Variety 174 A Local Criterion for Isomorphism 177 Projective Tangent Spaces 181 Determinantal Varieties 184 Lecture 15 Gauss Maps, Tangential and Dual Varieties 186 A Note About Characteristic 186 Gauss Maps 188 Tangential Varieties 189 The Variety of Tangent Lines 190 Joins of Intersecting Varieties 193 The Locus of Bitangent Lines 195 Dual Varieties 196 Lecture 16 Tangent Spaces to Grassmannians 200 Tangent Spaces to Grassmannians 200 Tangent Spaces to Incidence Correspondences 202 Varieties of Incident Planes 203 The Variety of Secant Lines 204 Varieties Swept out by Linear Spaces 204 The Resolution of the Generic Determinantal Variety 206 Tangent Spaces to Dual Varieties 208 Tangent Spaces to Fano Varieties 209 xviii Contents Lecture 17 Further Topics Involving Smoothness and Tangent Spaces 211 Gauss Maps on Curves 211 Osculating Planes and Associated Maps 213 The Second Fundamental Form 214 The Locus of Tangent Lines to a Variety 215 Bertini s Theorem 216 Blow-ups, Nash Blow-ups, and the Resolution of Singularities 219 Subadditivity of Codimensions of Intersections 222 Lecture 18 Degree 224 Bézouťs Theorem 227 The Rational Normal Curves 229 More Examples of Degrees 231 Veronese Varieties . 231 Segre Varieties 233 Degrees of Cones and Projections 234 Joins of Varieties 235 Unirationality of Cubic Hypersurfaces 237 Lecture 19 Further Examples and Applications of Degree 239 Multidegree of a Subvariety of a Product 239 Projective Degree of a Map 240 Joins of Corresponding Points 241 Varieties of Minimal Degree 242 Degrees of Determinantal Varieties 243 Degrees of Varieties Swept out by Linear Spaces 244 Degrees of Some Grassmannians 245 Harnack s Theorem 247 Lecture 20 Singular Points and Tangent Cones 251 Tangent Cones 251 Tangent Cones to Determinantal Varieties 256 Multiplicity 258 Examples of Singularities 260 Resolution of Singularities for Curves 264 Lecture 21 Parameter Spaces and Moduli Spaces 266 Parameter Spaces 266 Chow Varieties 268 Hubert Varieties 273 Contents xix Curves of Degree 2 275 Moduli Spaces 278 Plane Cubics 279 Lecture 22 Quadrics 282 Generalities about Quadrics 282 Tangent Spaces to Quadrics 283 Plane Conies 284 Quadric Surfaces 285 Quadrics in P 287 Linear Spaces on Quadrics 289 Lines on Quadrics 290 Planes on Four-Dimensional Quadrics 291 Fano Varieties of Quadrics in General 293 Families of Quadrics 295 The Variety of Quadrics in Ρ 295 The Variety of Quadrics in P2 296 Complete Conies 297 Quadrics in P 299 Pencils of Quadrics 301 Hints for Selected Exercises 308 References 314 Index 317
adam_txt	Contents Preface vii Acknowledgments ix Using This Book xi Part i: Examples of Varieties and Maps Lecture 1 Affine and Projective Varieties 3 A Note About Our Field 3 Affine Space and Affine Varieties 3 Projective Space and Projective Varieties 3 Linear Spaces 5 Finite Sets 6 Hypersurfaces 8 Analytic Subvarieties and Submanifolds 8 The Twisted Cubic 9 Rational Normal Curves 10 Determinantal Representation of the Rational Normal Curve 11 Another Parametrization of the Rational Normal Curve 11 The Family of Plane Conies 12 A Synthetic Construction of the Rational Normal Curve 13 Other Rational Curves 14 Varieties Defined over Subfields of К 16 A Note on Dimension, Smoothness, and Degree 16 Lecture 2 Regular Functions and Maps 17 The Zariski Topology 17 Regular Functions on an Affine Variety 18 Contents Projective Varieties 20 Regular Maps 21 The Veronese Map 23 Determinantal Representation of Veronese Varieties 24 Subvarieties of Veronese Varieties 24 The Segre Maps 25 Subvarieties of Segre Varieties 27 Products of Varieties 28 Graphs 29 Fiber Products 30 Combinations of Veronese and Segre Maps 30 Lecture 3 Cones, Projections, and More About Products 32 Cones 32 Quadrics 33 Projections 34 More Cones 37 More Projections 38 Constructible Sets 39 Lecture 4 Families and Parameter Spaces 41 Families of Varieties 41 The Universal Hyperplane 42 The Universal Hyperplane Section 43 Parameter Spaces of Hypersurfaces 44 Universal Families of Hypersurfaces 45 A Family of Lines 47 Lecture 5 Ideals of Varieties, Irreducible Decomposition, and the Nullstellensatz 48 Generating Ideals 48 Ideals of Projective Varieties 50 Irreducible Varieties and Irreducible Decomposition 51 General Objects 53 General Projections 54 General Twisted Cubics 55 Double Point Loci 56 A Little Algebra 57 Restatements and Corollaries 60 Lecture 6 Grassmannians and Related Varieties 63 Grassmannians 63 Subvarieties of Grassmannians 66 Contents xv The Grassmannian G(l, 3) 67 An Analog of the Veronese Map 68 Incidence Correspondences 68 Varieties of Incident Planes 69 The Join of Two Varieties 70 Fano Varieties 70 Lecture 7 Rational Functions and Rational Maps 72 Rational Functions 72 Rational Maps 73 Graphs of Rational Maps 75 Birational Isomorphism 77 The Quadric Surface 78 Hypersurfaces 79 Degree of a Rational Map 79 BIow-Ups 80 Blowing Up Points 81 Blowing Up Subvarieties 82 The Quadric Surface Again 84 The Cubic Scroll in P4 85 Unirationality 87 Lecture 8 More Examples 88 The Join of Two Varieties 88 The Secant Plane Map 89 Secant Varieties 90 Trisecant Lines, etc. 90 Joins of Corresponding Points 91 Rational Normal Scrolls 92 Higher-Dimensional Scrolls 93 More Incidence Correspondences 94 Flag Manifolds 95 More Joins and Intersections 95 Quadrics of Rank 4 96 Rational Normal Scrolls II 97 Lecture 9 Determinantal Varieties 98 Generic Determinantal Varieties 98 Segre Varieties 98 Secant Varieties of Segre Varieties 99 Linear Determinantal Varieties in General 99 Rational Normal Curves 100 Secant Varieties to Rational Normal Curves 103 Rational Normal Scrolls III Ю5 Contents Rational Normal Scrolls IV 109 More General Determinantal Varieties 111 Symmetrie and Skew-Symmetric Determinantal Varieties 112 Fano Varieties of Determinantal Varieties 112 Lecture 10 Algebraic Groups 114 The General Linear Group GL„K 114 The Orthogonal Group SO„K 115 The Symplectic Group Sp2nX 116 Group Actions 116 PGL.+jKactsonP" 116 PGL2X Acts on P2 117 PGL2K Acts on P3 118 PGL2K Acts on P" 119 PGL3K Acts on P5 120 PGL3K Acts on P9 121 РО„К Acts on P"~' (automorphisms of the Grassmannian) 122 PGL„K Acts on P(AkKn) 122 Quotients 123 Quotients of Affine Varieties by Finite Groups 124 Quotients of Affine Space 125 Symmetric Products 126 Quotients of Projective Varieties by Finite Groups 126 Weighted Projective Spaces 127 Part ii: Attributes of Varieties Lecture 11 Definitions of Dimension and Elementary Examples 133 Hypersurfaces 136 Complete Intersections 136 Immediate Examples 138 The Universal fc-Plane 142 Varieties of Incident Planes 142 Secant Varieties 143 Secant Varieties in General 146 Joins of Varieties 148 Flag Manifolds 148 (Some) Schubert Varieties 149 Lecture 12 M ore Dimension Computations 151 Determinantal Varieties 151 Fano Varieties 152 Parameter Spaces of Twisted Cubics 155 Twisted Cubics 155 Contents Twisted Cubics on a General Surface 156 Complete Intersections 157 Curves of Type (a, b) on а Quadric 158 Determinantal Varieties 159 Group Actions 161 GLtKlActsonSynťVandA^K 161 PGL^, К Acts on (Ρ")' and G(k, «)' 161 Lecture 13 Hubert Polynomials 163 Hubert Functions and Polynomials 163 Hubert Function of the Rational Normal Curve 166 Hubert Function of the Veronese Variety 166 Hubert Polynomials of Curves 166 Syzygies 168 Three Points in P2 170 Four Points in P2 171 Complete Intersections: Koszul Complexes 172 Lecture 14 Smoothness and Tangent Spaces 174 The Zariski Tangent Space to a Variety 174 A Local Criterion for Isomorphism 177 Projective Tangent Spaces 181 Determinantal Varieties 184 Lecture 15 Gauss Maps, Tangential and Dual Varieties 186 A Note About Characteristic 186 Gauss Maps 188 Tangential Varieties 189 The Variety of Tangent Lines 190 Joins of Intersecting Varieties 193 The Locus of Bitangent Lines 195 Dual Varieties 196 Lecture 16 Tangent Spaces to Grassmannians 200 Tangent Spaces to Grassmannians 200 Tangent Spaces to Incidence Correspondences 202 Varieties of Incident Planes 203 The Variety of Secant Lines 204 Varieties Swept out by Linear Spaces 204 The Resolution of the Generic Determinantal Variety 206 Tangent Spaces to Dual Varieties 208 Tangent Spaces to Fano Varieties 209 xviii Contents Lecture 17 Further Topics Involving Smoothness and Tangent Spaces 211 Gauss Maps on Curves 211 Osculating Planes and Associated Maps 213 The Second Fundamental Form 214 The Locus of Tangent Lines to a Variety 215 Bertini's Theorem 216 Blow-ups, Nash Blow-ups, and the Resolution of Singularities 219 Subadditivity of Codimensions of Intersections 222 Lecture 18 Degree 224 Bézouťs Theorem 227 The Rational Normal Curves 229 More Examples of Degrees 231 Veronese Varieties . 231 Segre Varieties 233 Degrees of Cones and Projections 234 Joins of Varieties 235 Unirationality of Cubic Hypersurfaces 237 Lecture 19 Further Examples and Applications of Degree 239 Multidegree of a Subvariety of a Product 239 Projective Degree of a Map 240 Joins of Corresponding Points 241 Varieties of Minimal Degree 242 Degrees of Determinantal Varieties 243 Degrees of Varieties Swept out by Linear Spaces 244 Degrees of Some Grassmannians 245 Harnack's Theorem 247 Lecture 20 Singular Points and Tangent Cones 251 Tangent Cones 251 Tangent Cones to Determinantal Varieties 256 Multiplicity 258 Examples of Singularities 260 Resolution of Singularities for Curves 264 Lecture 21 Parameter Spaces and Moduli Spaces 266 Parameter Spaces 266 Chow Varieties 268 Hubert Varieties 273 Contents xix Curves of Degree 2 275 Moduli Spaces 278 Plane Cubics 279 Lecture 22 Quadrics 282 Generalities about Quadrics 282 Tangent Spaces to Quadrics 283 Plane Conies 284 Quadric Surfaces 285 Quadrics in P" 287 Linear Spaces on Quadrics 289 Lines on Quadrics 290 Planes on Four-Dimensional Quadrics 291 Fano Varieties of Quadrics in General 293 Families of Quadrics 295 The Variety of Quadrics in Ρ ' 295 The Variety of Quadrics in P2 296 Complete Conies 297 Quadrics in P" 299 Pencils of Quadrics 301 Hints for Selected Exercises 308 References 314 Index 317
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spelling	Harris, Joe 1951- Verfasser (DE-588)112574718 aut Algebraic geometry a first course Joe Harris [reprint] New York [u.a.] Springer [2003] XIX, 328 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 133 Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s DE-604 Graduate texts in mathematics 133 (DE-604)BV000000067 133 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016392621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Harris, Joe 1951- Algebraic geometry a first course Graduate texts in mathematics Algebraische Geometrie (DE-588)4001161-6 gnd
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title	Algebraic geometry a first course
title_auth	Algebraic geometry a first course
title_exact_search	Algebraic geometry a first course
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title_short	Algebraic geometry
title_sort	algebraic geometry a first course
title_sub	a first course
topic	Algebraische Geometrie (DE-588)4001161-6 gnd
topic_facet	Algebraische Geometrie
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