Elements of the Theory of Algebraic Curves

Title......Page 1
Preface......Page 3
CONTENTS......Page 6
List of Symbols......Page 8
1 Introductory Remarks......Page 9
2 Homogeneous Coordinates......Page 14
Projective planes......Page 15
The line at infinity......Page 16
Projective planes over an extension field of K......Page 17
3 Algebraic Curves......Page 18
The equations of a curve......Page 19
Intersections......Page 23
Affine algebraic curves......Page 25
4 Resultants......Page 29
5 Linear Transformations......Page 35
Coordinate systems......Page 37
Transformations in the affine plane......Page 38
6 Simple Points and Singular Points......Page 39
The polar......Page 46
Some computational rules......Page 47
7 Bezout's Theorem......Page 51
8 The Basic Inequality......Page 56
9 Cubics......Page 63
Rational functions on an irreducible curve......Page 68
Abelian varieties......Page 73
Points of inflection......Page 75
Cubics through eight points......Page 81
The modulus of a nonsingular cubic......Page 83
11 Expansion at a Simple Point......Page 90
Formal power series......Page 91
12 Branches......Page 97
Imprimitive branch representations......Page 109
13 Generic Points......Page 111
Rational mappings......Page 113
Places......Page 119
14 Zeros and Poles......Page 121
Differentials......Page 123
The genus......Page 126
Infinitely near points......Page 135
15 Noetherian Conditions......Page 137
16 Linear Series......Page 143
17 Linear Systems......Page 151
18 The Theorem of Riemann-Roch......Page 155
Extension and reformulation of the theorem of Riemann-Roch......Page 157
19 The Question of the Characteristic......Page 160
20 Linear Series and Rational Mappings; Space Curves......Page 168
21 Analytic Branches......Page 181
22 Noetherian Conditions and Intersection Multiplicity......Page 194
23 Specialization; Space Curves......Page 201
Specializations and residue maps ("places")......Page 204
Valuations and morphisms ("places")......Page 207
24 Infinitely Near Points......Page 211
Notes......Page 217
Index......Page 221