Table of contents for Algebraic geometry : an introduction / Daniel Perrin ; translated from the French by Catriona Maclean.


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Introduction  ........................................  ........  1
0   Algebraic geometry  ................  ..................  1
1   Some objects .......................................
2   Some problems ..................................... 4
I    Affine algebraic sets ..................................... 9
1   Affine algebraic sets and the Zariski topology .............. 9
2   Ideal of an  affine algebraic set ............................  11
3   Irreducibility ...........................................  13
4   The Nullstellensatz (or Hilbert's zeros theorem) ............ 15
5   A first step towards B1zout's theorem ..................... 19
6   An introduction to morphisms ........................... 20
Exercises .................  ...........      ........... 24
II   Projective algebraic sets ................................. 25
0   Motivation ........................................     25
1   Projective space ...................................... ..25
2   Homographies ...................................... 27
3   Relation between affine and projective space ............... 27
4   Projective algebraic sets ................................ 29
5   Ideal of a  projective algebraic set .........................  31
6   A graded ring associated to a projective algebraic set ....... 32
7   Appendix: graded rings ................................. 33
Exercises ...........     .............................     34
III  Sheaves and varieties ..................................... 37
0    Motivation ........................................     37
1   The sheaf concept .................................. 38
2    The structural sheaf of an affine algebraic set .............. 41
3    Affine varieties ...................................... ...43
4    Algebraic varieties ......................................  44
5    Local rings ............................................ 47
6    Sheaves of modules  .....................................  48
7    Sheaves of modules on an affine algebraic variety ........... 50
8    Projective  varieties  .....................................  52
9    Sheaves of modules on projective algebraic varieties ......... 56
10  Two important exact sequences .......................... 59
11  Examples of morphisms ................................ 60
Exercises A ........................................... 63
Exercises B  ...............................................  66
IV   Dimension ................................................ 69
0    Introduction  ............... ............................  69
1   The topological definition and the link with algebra ......... 69
2    Dimension and counting equations ........................ 72
3    Morphisms and dimension ............................ 77
4    Annex: finite morphisms ................................. 82
Exercises ..........     ..............................      83
V    Tangent spaces and singular points ....................... 87
0    Introduction ....................................... 87
1   Tangent spaces ...................................... ...88
2    Singular points ...................................... ...91
3    Regular local rings ...................................... 93
4    Curves ................................................ 95
Exercises ........................................ 97
VI   B4zout's theorem ....................................101
0    Introduction ...........................................101
1   Intersection multiplicities .............................. 101
2    B6zout's theorem ..............  .................... 106
Exercises  ................................................ 111
VII Sheaf cohomology ...................................... .. 113
0    Introduction ...........    ...........................113
1   Some homological algebra ............................  114
2    Cech  cohomology  ....................................... 117
3    Vanishing  theorems ....................... ............. 121
4    The cohomology of the sheaves Opn (d) .................. 122
Exercises ..............................................127
VIII Arithmetic genus of curves and the weak Riemann-Roch
theorem   ...........................   ..................131
0    Introduction: the Euler-Poincar6 characteristic ............. 131
1   Degree and genus of projective curves, Riemann-Roch 1 ..... 132
2    Divisors on a curve and Riemann-Roch 2 .................. 138
Exercises .............................................147
IX   Rational maps, geometric genus and rational curves ...... 149
0    Introduction ......................................149
1    Rational maps ....................................149
2    Curves  ................................................152
3    Normalisation: the algebraic method .....................155
4    Affine blow-ups  ........................................158
5    Global blow-ups ...................................163
6    Appendix: review of the above proofs ..................... 170
X    Liaison of space curves ..........    .................... 173
0    Introduction ...........................................173
1   Ideals and resolutions ..................................174
2    ACM curves ..........     ............................179
3    Liaison of space curves ................................. 187
Exercises .............   ..................................194
Appendices
A    Summary of useful results from algebra ................... 199
1    Rings ................................................199
2    Tensor products ........................................204
3    Transcendence bases .................................... 206
4    Some algebra exercises .................................. 207
B    Schemes ...............    .................................209
0    Introduction ......................................209
1   Affine schemes .........................................210
2    Schemes   ...................................210
3    What changes when we work with schemes ................ 211
4    Why working with schemes is useful ................... .. 212
5    A scheme-theoretic Bertini theorem ....................... 213
C    Problems .................................................. 215
Problem I ................................................... 215
Problem II ............................................     217
Problem  III .................................................218
Problem IV ..........................................220
Problem V ..................................................222
Problem VI ...........     ...............................223
Problem VII ...............................................225
Problem VIII .........................................228
Problem IX .................................................230
Midterm, December 1991 ................................... 232
Exam, January 1992 .......................................234
Exam, June 1992 ......................................238
Exam, January 1993 .......................................239
Exam, June 1993 ......................................242
Exam, February 1994 ...................................245



Library of Congress subject headings for this publication: Geometry, Algebraic Textbooks