## 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

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