Selected topics in algebraic geometry:
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
New York
Chelsea Publ.
1970
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 484 S. |
| ISBN: | 0828401896 |
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| 245 | 1 | 0 | |a Selected topics in algebraic geometry |c by Virgil Snyder ... |
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| adam_text | CONTENTS OF VOLUME I
PACE
List of Periodicals and Abbreviations H
Books 18a
CHAPTER 1
QUADRATIC CREMONA TRANSFORMATIONS
Arnold Emck
A. Early Examples of Quadratic Transformations 19
B. General Quadratic Transformation 23
C. Special Quadratic Transformations 28
1. The involutorial quadratic transformation 28
2. Quadric inversion 30
3. Special transformations obtained by various geometric devices 30
D. Inversion. Circle Transformation 34
1. Introduction 34
2. Inversion 35
3. Circle transformation 39
4. Geometry of the circle and the sphere 42
E. Various Applications. Groups of Quadratic Transformations 42
1. Geometric applications. Invariant algebraic curves 42
2. Other curves associated with quadratic Cremona transformations... 44
3. Repeated quadratic transformations 45
4. Groups of quadratic transformations 46
P. Systems of Correlations and Collineations in Connection with Quadratic
Cremona Transformations 47
1. Hirst s investigation on the correlation of two planes 47
2. Further investigations on correlation and collineations 48
3. Trilinear correspondence. Apolarity 49
Bibliography 50
CHAPTER II
ANALYSIS OF SINGULARITIES OF PLANE ALGEBRAIC CURVES
Arnold Emch
1. Account of the development of the theory of singularities of plane alge¬
braic curves 56
2. Consecutive points of a singularity; characteristics. Birational transfor¬
mations of plane algebraic curves into non singular space curves 65
3. Multiplicities of intersections; additional references on singularities 68
Bibliography 70
1
2 CONTENTS
CHAPTER III
LINEAR SYSTEMS OF PLANE CURVES
F. R. Sharps
page
1. Definitions 75
2. Superabundance 76
3. Characteristic series 76
4. Maximum dimensions 76
5. Fundamental curve 77
6. Adjoints 77
7. Virtual characteristics of composite system 77
8. Adjoints and dimensions 77
Bibliography 78
CHAPTER IV
PLANAR CREMONA TRANSFORMATIONS
Arthur B. Coble
1. Definitions and general properties 79
2. Determination of the types 82
3. Linear transformations with integer or rational coefficients associated with
Cremona transformations. The geometry of the Cremona group 85
4. Construction of the F Points. Construction and description of particular
transformations 89
5. Isologous curves. Fixed points and curves. Cyclic sets 92
6. Involutorial cremona transformations 94
7. The reduction of linear systems of curves 98
8. Periodic transformations and finite groups 102
9. Infinite discontinuous groups 107
10. Continuous groups 107
11. Geometric applications 109
12. Algebraic and other applications 113
Bibliography 114
CHAPTER V
MULTIPLE CORRESPONDENCES BETWEEN TWO PLANES
Virgil Snyder
1. The (1, 2) correspondences between S and 122
2. Fundamental elements 124
3. Derivation of the types 125
4. (1, A;) correspondences between two planes, k~ 2 128
5. Cyclic involutions in 133
6. Multiple correspondences (m, n) between two planes 133
Bibliography 137
.¦¦ ¦*•• •..•
CONTENTS 3
CHAPTER VI
INVOLUTIONS ON RATIONAL CURVES
Virgil Snydeb
page
1. Involutions on Ci 140
2. Involutions on a conic 141
3. General properties of In , independent of carrier 143
4. Plane rational cubic curves 145
5. Space cubic curves and various plane mapping 147
6. Plane rational quartics 149
7. Rational quartics in 150
8. Plane curves of order higher than 4 153
9. Rational curves C» (n 4) in , fc 2 154
10. Rational involutions In on rational carriers 157
Bibliography 160
CHAPTER VII
CORRESPONDENCES ON NON RATIONAL CURVES
Virgil Snydbr
1. Chasles s principle of correspondence 166
2. Inscribed and Steiner polygons 168
3. Rational involutions on curves 169
4. Involutions of higher dimensions 170
5. Rational involutions on curves of genus p 0 171
6. Cayley Brill principle of correspondence 173
7. Singular correspondences 174
8. Zeuthen s formula 174
9. Correspondence on a cubic curve 175
10. Generation of cubic curves 175
11. Quartic curves 176
12. Contact conies of quartics 177
13. The Brill Noether theory. The fundamental theorem 178
14. Linear series of point groups 178
15. Normal curves 180
16. Reduction to forms with double points 180
17. Particular curves of order n 4 181
18. Irrational involutions 182
19. Relations to enumerative geometry 184
20. Applications of binary forms 186
21. Polar properties 187
Bibliography !87
4 CONTENTS
CHAPTER VIII
CREMONA TRANSFORMATIONS IN SPACE AND HYPERSPACE
Arthur B. Coble
page
1. Definitions and general properties I97
2. Determination of the types 200
3. Integer linear transformations associated with regular groups 205
4. Projective descriptions of particular types 208
5. Fixed points and cyclic sets. The complex f 211
6. Periodic transformations and finite groups 212
7. Infinite discontinuous groups 214
8. Continuous groups 216
9. Geometric applications 217
10. Algebraic and other applications 220
Bibliography 221
CHAPTER IX
(1, 2) CORRESPONDENCES BETWEEN S r AND Sr, r 2
Virgil Snyder
1. Reciprocal radii 227
2. Involutorial transformations of space 228
3. The (1, 2) correspondences between two spaces S s, Ss 231
4. Fundamental elements in the two spaces 234
5. Various particular cases 235
6. (1, 2) correspondences between and S 237
7. Involutions of lines in 238
8. Line complexes and congruences associated with I, 241
9. Infinite discontinuous birational groups 245
10. Involutions h belonging to M3 in 247
Bibliography 248
CHAPTER X
REDUCTION OF SINGULARITIES OF SPACE CURVES AND SURFACES
Arnold Emch
1. Introduction 252
2. Singularities of space curves 252
3. Singularities of surfaces 252
4. Reduction of singularities 253
Bibliography 254
CONTENTS 5
CHAPTER XI
MULTIPLE CORRESPONDENCE IN SPACE AND HYPERSPACE
Virgil Snyder
PAGE
1. (1, k) correspondences between S3 S s, (fc 2) 257
2. Lines of complexes mapped on points of 259
3. Lines of mapped on points of 263
4. Linear line complexes mapped on other elements 265
5. Various correspondences in hyperspace 266
6. Compound involutions 269
Bibliography 270
CHAPTER XII
THE MAPPING OF A RATIONAL SURFACE ON A PLANE
F. R. Shabpe
1. The quadric surface 275
2. The cubic surface 275
3. Quartic surface with a double line 276
4. Ft with a double conic 276
5. The Steiner surface 277
6. Rational quartics with no double curve 277
7. Quintics with 2 double skew lines 278
8. Quintics with a double cubic 278
9. Quintics with double C, p= 278
10. Quintics with double quintic with a triple point 279
11. Quintics with singular points 279
12. Rational Fn n 5 279
13. Fn with an n 2 fold line 280
14. Fn with an n 3 fold line and n 3 triple points 280
15. Rational ruled surfaces 28°
16. i?4 with a double cubic 28°
17. Rn with an n 1 fold line 2§1
18. General theory of mapping a rational Fn on a plane 281
19. Surfaces in derived by projection from a higher space 282
20. Miscellaneous methods 283
21. Rational surfaces and linear systems of curves 284
22. Rational surfaces mappable on a multiple plane 284
23. General conditions for the rationality of a surface 285
24. Rational surfaces with plane sections of genus 3 286
25. Rational surface containing an infinite number of conies 287
Bibliography 287
6 CONTENTS
CHAPTER XIII
THE MAPPING OP A RATIONAL CONGRUENCE ON A PLANE
F. R. Shabpe
PAGE
1. Introduction 291
2. Congruences with a finite number of singular points 291
3. Congruences with a curve of singular points 292
4. Congruences defined by quadrics and projectivities 293
Bibliography 294
CHAPTER XIV
INVOLUTIONS ON IRRATIONAL SURFACES
Charles H. Sisam
1. Definitions 295
I. Involutions with a Curve of Coincidences 295
2. Relations between invariants 295
3. The double plane 296
4. Some special double planes 297
5. The cyclic multiple plane 298
6. Other special surfaces 298
II. Involutions Having a Finite Number of Coincidences 299
7. General theorems 299
8. Involutions of given order 300
9. Involutions of genera one on surfaces of genera one 302
10. Involutions of genera pa=Pa=0, P2=1 on surfaces of genera one 304
11. Involutions of genera pa—Pa=0, / 2=l on surfaces of the same
genera 305
12. Involutions on other special surfaces 306
Bibliography 308
CHAPTER XV
TRANSCENDENTAL THEORY
S. Lefschetz
1. Algebraic curves 310
2. Linear systems of curves on F and related invariants 315
3. Topology of F 317
4. Integrals of F 321
5. Distribution of the curves of F and the bases 322
6. Extension to Vr 327
Glossary of new terms and notations 328
Bibliography 329
CONTENTS 7
CHAPTER XVI
SINGULAR CORRESPONDENCES BETWEEN ALGEBRAIC CURVES
S. Lefschetz
PAGE
1. Generalities 331
2. Singular correspondence on C 331
3. Correspondences between Cp and C« 338
4. Irrational involutions and related topics 340
5. Irrational series 342
6. Birational transformations 344
Bibliography 346
CHAPTER XVII
HYPERELLIPTIC SURFACES AND ABELIAN VARIETIES
S. Lefschetz
1. Multiply periodic and related functions 349
2. Hyperelliptic surfaces 356
3. Abelian varieties. General properties 369
4. Impure matrices and their varieties 375
5. Transformations of Vp 378
6. Complex multiplication 38°
7. Additional topics 389
Bibliography 392
|
| adam_txt |
CONTENTS OF VOLUME I
PACE
List of Periodicals and Abbreviations H
Books 18a
CHAPTER 1
QUADRATIC CREMONA TRANSFORMATIONS
Arnold Emck
A. Early Examples of Quadratic Transformations 19
B. General Quadratic Transformation 23
C. Special Quadratic Transformations 28
1. The involutorial quadratic transformation 28
2. Quadric inversion 30
3. Special transformations obtained by various geometric devices 30
D. Inversion. Circle Transformation 34
1. Introduction 34
2. Inversion 35
3. Circle transformation 39
4. Geometry of the circle and the sphere 42
E. Various Applications. Groups of Quadratic Transformations 42
1. Geometric applications. Invariant algebraic curves 42
2. Other curves associated with quadratic Cremona transformations. 44
3. Repeated quadratic transformations 45
4. Groups of quadratic transformations 46
P. Systems of Correlations and Collineations in Connection with Quadratic
Cremona Transformations 47
1. Hirst's investigation on the correlation of two planes 47
2. Further investigations on correlation and collineations 48
3. Trilinear correspondence. Apolarity 49
Bibliography 50
CHAPTER II
ANALYSIS OF SINGULARITIES OF PLANE ALGEBRAIC CURVES
Arnold Emch
1. Account of the development of the theory of singularities of plane alge¬
braic curves 56
2. Consecutive points of a singularity; characteristics. Birational transfor¬
mations of plane algebraic curves into non singular space curves 65
3. Multiplicities of intersections; additional references on singularities 68
Bibliography 70
1
2 CONTENTS
CHAPTER III
LINEAR SYSTEMS OF PLANE CURVES
F. R. Sharps
page
1. Definitions 75
2. Superabundance 76
3. Characteristic series 76
4. Maximum dimensions 76
5. Fundamental curve 77
6. Adjoints 77
7. Virtual characteristics of composite system 77
8. Adjoints and dimensions 77
Bibliography 78
CHAPTER IV
PLANAR CREMONA TRANSFORMATIONS
Arthur B. Coble
1. Definitions and general properties 79
2. Determination of the types 82
3. Linear transformations with integer or rational coefficients associated with
Cremona transformations. The geometry of the Cremona group 85
4. Construction of the F Points. Construction and description of particular
transformations 89
5. Isologous curves. Fixed points and curves. Cyclic sets 92
6. Involutorial cremona transformations 94
7. The reduction of linear systems of curves 98
8. Periodic transformations and finite groups 102
9. Infinite discontinuous groups 107
10. Continuous groups 107
11. Geometric applications 109
12. Algebraic and other applications 113
Bibliography 114
CHAPTER V
MULTIPLE CORRESPONDENCES BETWEEN TWO PLANES
Virgil Snyder
1. The (1, 2) correspondences between S\ and 122
2. Fundamental elements 124
3. Derivation of the types 125
4. (1, A;) correspondences between two planes, k~ 2 128
5. Cyclic involutions in 133
6. Multiple correspondences (m, n) between two planes 133
Bibliography 137
.¦¦ '¦*•• •.•
CONTENTS 3
CHAPTER VI
INVOLUTIONS ON RATIONAL CURVES
Virgil Snydeb
page
1. Involutions on Ci 140
2. Involutions on a conic 141
3. General properties of In', independent of carrier 143
4. Plane rational cubic curves 145
5. Space cubic curves and various plane mapping 147
6. Plane rational quartics 149
7. Rational quartics in 150
8. Plane curves of order higher than 4 153
9. Rational curves C» (n 4) in , fc 2 154
10. Rational involutions In" on rational carriers 157
Bibliography 160
CHAPTER VII
CORRESPONDENCES ON NON RATIONAL CURVES
Virgil Snydbr
1. Chasles's principle of correspondence 166
2. Inscribed and Steiner polygons 168
3. Rational involutions on curves 169
4. Involutions of higher dimensions 170
5. Rational involutions on curves of genus p 0 171
6. Cayley Brill principle of correspondence 173
7. Singular correspondences 174
8. Zeuthen's formula 174
9. Correspondence on a cubic curve 175
10. Generation of cubic curves 175
11. Quartic curves 176
12. Contact conies of quartics 177
13. The Brill Noether theory. The fundamental theorem 178
14. Linear series of point groups 178
15. Normal curves 180
16. Reduction to forms with double points 180
17. Particular curves of order n 4 181
18. Irrational involutions 182
19. Relations to enumerative geometry 184
20. Applications of binary forms 186
21. Polar properties 187
Bibliography !87
4 CONTENTS
CHAPTER VIII
CREMONA TRANSFORMATIONS IN SPACE AND HYPERSPACE
Arthur B. Coble
page
1. Definitions and general properties I97
2. Determination of the types 200
3. Integer linear transformations associated with regular groups 205
4. Projective descriptions of particular types 208
5. Fixed points and cyclic sets. The complex f 211
6. Periodic transformations and finite groups 212
7. Infinite discontinuous groups 214
8. Continuous groups 216
9. Geometric applications 217
10. Algebraic and other applications 220
Bibliography 221
CHAPTER IX
(1, 2) CORRESPONDENCES BETWEEN S'r AND Sr, r 2
Virgil Snyder
1. Reciprocal radii 227
2. Involutorial transformations of space 228
3. The (1, 2) correspondences between two spaces S's, Ss 231
4. Fundamental elements in the two spaces 234
5. Various particular cases 235
6. (1, 2) correspondences between and S\ 237
7. Involutions of lines in 238
8. Line complexes and congruences associated with I, 241
9. Infinite discontinuous birational groups 245
10. Involutions h belonging to M3 in 247
Bibliography 248
CHAPTER X
REDUCTION OF SINGULARITIES OF SPACE CURVES AND SURFACES
Arnold Emch
1. Introduction 252
2. Singularities of space curves 252
3. Singularities of surfaces 252
4. Reduction of singularities 253
Bibliography 254
CONTENTS 5
CHAPTER XI
MULTIPLE CORRESPONDENCE IN SPACE AND HYPERSPACE
Virgil Snyder
PAGE
1. (1, k) correspondences between S3 S's, (fc 2) 257
2. Lines of complexes mapped on points of 259
3. Lines of mapped on points of 263
4. Linear line complexes mapped on other elements 265
5. Various correspondences in hyperspace 266
6. Compound involutions 269
Bibliography 270
CHAPTER XII
THE MAPPING OF A RATIONAL SURFACE ON A PLANE
F. R. Shabpe
1. The quadric surface 275
2. The cubic surface 275
3. Quartic surface with a double line 276
4. Ft with a double conic 276
5. The Steiner surface 277
6. Rational quartics with no double curve 277
7. Quintics with 2 double skew lines 278
8. Quintics with a double cubic 278
9. Quintics with double C, p=\ 278
10. Quintics with double quintic with a triple point 279
11. Quintics with singular points 279
12. Rational Fn n 5 279
13. Fn with an n 2 fold line 280
14. Fn with an n 3 fold line and n 3 triple points 280
15. Rational ruled surfaces 28°
16. i?4 with a double cubic 28°
17. Rn with an n 1 fold line 2§1
18. General theory of mapping a rational Fn on a plane 281
19. Surfaces in derived by projection from a higher space 282
20. Miscellaneous methods 283
21. Rational surfaces and linear systems of curves 284
22. Rational surfaces mappable on a multiple plane 284
23. General conditions for the rationality of a surface 285
24. Rational surfaces with plane sections of genus 3 286
25. Rational surface containing an infinite number of conies 287
Bibliography 287
6 CONTENTS
CHAPTER XIII
THE MAPPING OP A RATIONAL CONGRUENCE ON A PLANE
F. R. Shabpe
PAGE
1. Introduction 291
2. Congruences with a finite number of singular points 291
3. Congruences with a curve of singular points 292
4. Congruences defined by quadrics and projectivities 293
Bibliography 294
CHAPTER XIV
INVOLUTIONS ON IRRATIONAL SURFACES
Charles H. Sisam
1. Definitions 295
I. Involutions with a Curve of Coincidences 295
2. Relations between invariants 295
3. The double plane 296
4. Some special double planes 297
5. The cyclic multiple plane 298
6. Other special surfaces 298
II. Involutions Having a Finite Number of Coincidences 299
7. General theorems 299
8. Involutions of given order 300
9. Involutions of genera one on surfaces of genera one 302
10. Involutions of genera pa=Pa=0, P2=1 on surfaces of genera one 304
11. Involutions of genera pa—Pa=0, / 2=l on surfaces of the same
genera 305
12. Involutions on other special surfaces 306
Bibliography 308
CHAPTER XV
TRANSCENDENTAL THEORY
S. Lefschetz
1. Algebraic curves 310
2. Linear systems of curves on F and related invariants 315
3. Topology of F 317
4. Integrals of F 321
5. Distribution of the curves of F and the bases 322
6. Extension to Vr 327
Glossary of new terms and notations 328
Bibliography 329
CONTENTS 7
CHAPTER XVI
SINGULAR CORRESPONDENCES BETWEEN ALGEBRAIC CURVES
S. Lefschetz
PAGE
1. Generalities 331
2. Singular correspondence on C 331
3. Correspondences between Cp and C« 338
4. Irrational involutions and related topics 340
5. Irrational series 342
6. Birational transformations 344
Bibliography 346
CHAPTER XVII
HYPERELLIPTIC SURFACES AND ABELIAN VARIETIES
S. Lefschetz
1. Multiply periodic and related functions 349
2. Hyperelliptic surfaces 356
3. Abelian varieties. General properties 369
4. Impure matrices and their varieties 375
5. Transformations of Vp 378
6. Complex multiplication 38°
7. Additional topics 389
Bibliography 392 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| id | DE-604.BV021929455 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:06:13Z |
| indexdate | 2024-07-09T20:47:37Z |
| institution | BVB |
| isbn | 0828401896 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015144612 |
| oclc_num | 468230081 |
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| owner | DE-706 DE-83 DE-11 |
| owner_facet | DE-706 DE-83 DE-11 |
| physical | 484 S. |
| publishDate | 1970 |
| publishDateSearch | 1970 |
| publishDateSort | 1970 |
| publisher | Chelsea Publ. |
| record_format | marc |
| spelling | Selected topics in algebraic geometry by Virgil Snyder ... 2. ed. New York Chelsea Publ. 1970 484 S. txt rdacontent n rdamedia nc rdacarrier Géométrie algébrique ram Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 s DE-604 Snyder, Virgil 1869-1950 Sonstige (DE-588)117442097 oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015144612&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Selected topics in algebraic geometry Géométrie algébrique ram Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4001161-6 |
| title | Selected topics in algebraic geometry |
| title_auth | Selected topics in algebraic geometry |
| title_exact_search | Selected topics in algebraic geometry |
| title_exact_search_txtP | Selected topics in algebraic geometry |
| title_full | Selected topics in algebraic geometry by Virgil Snyder ... |
| title_fullStr | Selected topics in algebraic geometry by Virgil Snyder ... |
| title_full_unstemmed | Selected topics in algebraic geometry by Virgil Snyder ... |
| title_short | Selected topics in algebraic geometry |
| title_sort | selected topics in algebraic geometry |
| topic | Géométrie algébrique ram Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Géométrie algébrique Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015144612&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT snydervirgil selectedtopicsinalgebraicgeometry |