Selected topics in algebraic geometry

Title page......Page 1
Date-line......Page 2
Title......Page 3
PREFACE TO VOLUME I......Page 5
PREFACE TO VOLUME II......Page 6
CONTENTS OF VOLUME I......Page 7
CONTENTS OF VOLUME II......Page 14
List of Periodicals and Abbreviations......Page 17
Books......Page 25
A. Early Examples of Quadratic Transformations......Page 29
B. General Quadratic Transformation......Page 33
1. The involutorial quadratic transformation......Page 38
3. Special transformations obtained by various geometric devices......Page 40
1. Introduction......Page 44
2. Inversion......Page 45
3. Circle transformation......Page 49
1. Geometric applications. Invariant algebraic curves......Page 52
2. Other curves associated with quadratic Cremona transformations......Page 54
3. Repeated quadratic transformations......Page 55
4. Groups of quadratic transformations......Page 56
1. Hirst's investigation on the correlation of two planes......Page 57
2. Further investigations on correlation and collineations......Page 58
3. Trilinear correspondence. Apolarity......Page 59
Bibliography......Page 60
1. Account of the development of the theory of singularities of plane algebraic curves......Page 66
2. Consecutive points of a singularity; characteristics. Birational transformations of plane algebraic curves into non-singular space curves......Page 75
3. Multiplicities of intersections; additional references on singularities......Page 78
Bibliography......Page 80
1. Definitions......Page 85
4. Maximum dimensions......Page 86
8. Adjoints and dimensions......Page 87
Bibliography......Page 88
1. Definitions and general properties......Page 89
2. Determination of the types......Page 92
3. Linear transformations with integer or rational coefficients associated with Cremona transformations. The geometry of the Cremona group......Page 95
4. Construction of the $F$-Points. Construction and description of particular transformations......Page 99
5. Isologous curves. Fixed points and curves. Cyclic sets......Page 102
6. Involutorial cremona transformations......Page 104
7. The reduction of linear systems of curves......Page 108
8. Periodic transformations and finite groups......Page 112
10. Continuous groups......Page 117
11. Geometric applications......Page 119
12. Algebraic and other applications......Page 123
Bibliography......Page 124
1. The $(1,2)$ correspondences between $S'_2$ and $S_2$......Page 132
2. Fundamental elements......Page 134
3. Derivation of the types......Page 135
4. $(1,k)$ correspondences between two planes, $k>2$......Page 138
6. Multiple correspondences $(m,n)$ between two planes......Page 143
Bibliography......Page 147
1. Involutions on $C_1$......Page 150
2. Involutions on a conic......Page 151
3. General properties of $I_n^1$, independent of carrier......Page 153
4. Plane rational cubic curves......Page 155
5. Space cubic curves and various plane mapping......Page 157
6. Plane rational quartics......Page 159
7. Rational quartics in $S_3$......Page 160
8. Plane curves of order higher than 4......Page 163
9. Rational curves $C_n$ $(n>4)$ in $S_k$, $k>2$......Page 164
10. Rational involutions $I_n^k$ on rational carriers......Page 167
Bibliography......Page 170
1. Chasles's principle of correspondence......Page 176
2. Inscribed and Steiner polygons......Page 178
3. Rational involutions on curves......Page 179
4. Involutions of higher dimensions......Page 180
5. Rational involutions on curves of genus $p>0$......Page 181
6. Cayley-Brill principle of correspondence......Page 183
8. Zeuthen's formula......Page 184
10. Generation of cubic curves......Page 185
11. Quartic curves......Page 186
12. Contact conies of quartics......Page 187
14. Linear series of point groups......Page 188
16. Reduction to forms with double points......Page 190
17. Particular curves of order $n>4$......Page 191
18. Irrational involutions......Page 192
19. Relations to enumerative geometry......Page 194
20. Applications of binary forms......Page 196
Bibliography......Page 197
1. Definitions and general properties......Page 207
2. Determination of the types......Page 210
3. Integer linear transformations associated with regular groups......Page 215
4. Projective descriptions of particular types......Page 218
5. Fixed points and cyclic sets. The complex $\Gamma$......Page 221
6. Periodic transformations and finite groups......Page 222
7. Infinite discontinuous groups......Page 224
8. Continuous groups......Page 226
9. Geometric applications......Page 227
10. Algebraic and other applications......Page 230
Bibliography......Page 231
1. Reciprocal radii......Page 237
2. Involutorial transformations of space......Page 238
3. The $(1,2)$ correspondences between two spaces $S'_3$, $S_3$......Page 241
4. Fundamental elements in the two spaces......Page 244
5. Various particular cases......Page 245
6. $(1,2)$ correspondences between $S_4$ and $S'_4$......Page 247
7. Involutions of lines in $S_3$......Page 248
8. Line complexes and congruences associated with $I_2$......Page 251
9. Infinite discontinuous birational groups......Page 255
10. Involutions $I_2$ belonging to $M_3$ in $S_4$......Page 257
Bibliography......Page 258
3. Singularities of surfaces......Page 262
4. Reduction of singularities......Page 263
Bibliography......Page 264
1. $(1,k)$ correspondences between $S_3$ $S'_3$, $(k>2)$......Page 267
2. Lines of complexes mapped on points of $S_3$......Page 269
3. Lines of $S_3$ mapped on points of $S_4$......Page 273
4. Linear line complexes mapped on other elements......Page 275
5. Various correspondences in hyperspace......Page 276
6. Compound involutions......Page 279
Bibliography......Page 280
2. The cubic surface......Page 285
4. $F_4$ with a double conic......Page 286
6. Rational quartics with no double curve......Page 287
9. Quintics with double $C_4$, $p=1$......Page 288
12. Rational $F_n$ $n>5$......Page 289
16. $R_4$ with a double cubic......Page 290
18. General theory of mapping a rational $F_n$ on a plane......Page 291
19. Surfaces in $S_3$ derived by projection from a higher space......Page 292
20. Miscellaneous methods......Page 293
22. Rational surfaces mappable on a multiple plane......Page 294
23. General conditions for the rationality of a surface......Page 295
24. Rational surfaces with plane sections of genus 3......Page 296
Bibliography......Page 297
2. Congruences with a finite number of singular points......Page 301
3. Congruences with a curve of singular points......Page 302
4. Congruences defined by quadrics and projectivities......Page 303
Bibliography......Page 304
2. Relations between invariants......Page 305
3. The double plane......Page 306
4. Some special double planes......Page 307
6. Other special surfaces......Page 308
7. General theorems......Page 309
8. Involutions of given order......Page 310
9. Involutions of genera one on surfaces of genera one......Page 312
10. Involutions of genera $p_a=P_3=0$, $P_2=1$ on surfaces of genera one......Page 314
11. Involutions of genera $p_a=P_3=0$, $P_2=1$ on surfaces of the same genera......Page 315
12. Involutions on other special surfaces......Page 316
Bibliography......Page 318
1. Algebraic curves......Page 320
2. Linear systems of curves on $F$ and related invariants......Page 325
3. Topology of $F$......Page 327
4. Integrals of $F$......Page 331
5. Distribution of the curves of $F$ and the bases......Page 332
6. Extension to $V_r$......Page 337
Glossary of new terms and notations......Page 338
Bibliography......Page 339
2. Singular correspondence on $C^p$......Page 341
3. Correspondences between $C^p$ and $C^q$......Page 348
4. Irrational involutions and related topics......Page 350
5. Irrational series......Page 352
6. Birational transformations......Page 354
Bibliography......Page 356
1. Multiply periodic and related functions......Page 359
2. Hyperelliptic surfaces......Page 366
3. Abelian varieties. General properties......Page 379
4. Impure matrices and their varieties......Page 385
5. Transformations of $V_p$......Page 388
6. Complex multiplication......Page 390
7. Additional topics......Page 399
Bibliography......Page 402
VOLUME II......Page 409
b) Quartics......Page 411
c) Curves of higher order......Page 412
a) General......Page 413
b) Cubics......Page 414
d) Various curves......Page 415
5. Trisecant surfaces of space curves......Page 416
6. Plane sextics with six cusps......Page 417
10. Incidence theorems......Page 418
11. Various theorems......Page 419
Bibliography......Page 420
1. Systems of lines......Page 424
2. Systems of conies......Page 425
3. Systems of cubics......Page 427
4. Systems of quartics......Page 429
5. Miscellaneous......Page 431
Bibliography......Page 433
1. Complexes......Page 435
2. Ruled surfaces and other systems......Page 437
3. Geometry of the sphere......Page 439
4. Surfaces and varieties; irregularity......Page 440
5. Involutions on surfaces......Page 441
6. Characteristics of surfaces......Page 442
7. Generation of curves, surfaces and varieties......Page 443
Bibliography......Page 444
1. Plane quadratic inversions......Page 447
2. Types......Page 448
4. Systems of cubic curves......Page 449
5. Finite groups......Page 450
6. Plane involutions......Page 451
7. Quadratic space transformations......Page 452
8. Hyperspace transformations......Page 453
10. Fundamental elements......Page 454
12. Tables of characteristics......Page 456
13. Series of composition......Page 457
14. Cremona transformations in hyperspace......Page 458
15. Invariants of surfaces and of varieties......Page 460
16. Involutorial transformations in $S_r$, $r\geq 3$......Page 461
17. Involutions denned by a $(k,1)$ correspondence......Page 462
18. Involutions on $V_3^3$ of $S_4$......Page 463
20. Cremona transformations connected with line complexes in $Sr$......Page 464
21. Cyclic involutions of order $n>2$......Page 465
23. Montesano's multiplier......Page 466
Bibliography......Page 467
1. $(1,n)$ correspondences......Page 473
3. Double planes......Page 474
4. Methods of generation......Page 475
6. Double fives......Page 477
8. Application to lines, spheres, etc......Page 478
9. Systems of curves......Page 479
Bibliography......Page 480
2. Quartics and ruled surfaces in $S_3$......Page 483
3. Quintic surfaces in $S_3$......Page 485
5. Configurations, surfaces and varieties in $S_n$......Page 486
Bibliography......Page 492