دانلود کتاب Algebraic Curves: An Introduction to Algebraic Geometry

فهرست مطالب :

Preface 1 Affine Algebraic Sets 1.1 Algebraic Preliminaries 1.2 Affine Space and Algebraic Sets 1.3 The Ideal of a Set of Points 1.4 The Hilbert Basis Theorem 1.5 Irreducible Components of an Algebraic Set 1.6 Algebraic Subsets of the Plane 1.7 Hilbert’s Nullstellensatz 1.8 Modules; Finiteness Conditions 1.9 Integral Elements 1.10 Field Extensions 2 Affine Varieties 2.1 Coordinate Rings 2.2 Polynomial Maps 2.3 Coordinate Changes 2.4 Rational Functions and Local Rings 2.5 Discrete Valuation Rings 2.6 Forms 2.7 Direct Products of Rings 2.8 Operations with Ideals 2.9 Ideals with a Finite Number of Zeros 2.10 Quotient Modules and Exact Sequences 2.11 Free Modules 3 Local Properties of Plane Curves 3.1 Multiple Points and Tangent Lines 3.2 Multiplicities and Local Rings 3.3 Intersection Numbers 4 Projective Varieties 4.1 Projective Space 4.2 Projective Algebraic Sets 4.3 Affine and Projective Varieties 4.4 Multiprojective Space 5 Projective Plane Curves 5.1 Definitions 5.2 Linear Systems of Curves 5.3 Bézout’s Theorem 5.4 Multiple Points 5.5 Max Noether’s Fundamental Theorem 5.6 Applications of Noether’s Theorem 6 Varieties, Morphisms, and Rational Maps 6.1 The Zariski Topology 6.2 Varieties 6.3 Morphisms of Varieties 6.4 Products and Graphs 6.5 Algebraic Function Fields and Dimension of Varieties 6.6 Rational Maps 7 Resolution of Singularities 7.1 Rational Maps of Curves 7.2 Blowing up a Point in A^2 7.3 Blowing up Points in P^2 7.4 Quadratic Transformations 7.5 Nonsingular Models of Curves 8 Riemann-Roch Theorem 8.1 Divisors 8.2 The Vector Spaces L(D) 8.3 Riemann’s Theorem 8.4 Derivations and Differentials 8.5 Canonical Divisors 8.6 Riemann-Roch Theorem Appendix A - Nonzero Characteristic Appendix B - Suggestions for Further Reading Appendix C - Notation