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Email to friend
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Algebraic Geometry For Beginners
C Musili
ISBN :
81-85931-27-5
Year Of Publication :
2001
Price : US$ 15
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This book offers a self-contained elementary introduction to the fundamental concepts and techniques of Algebraic Geometry, leading to some gems of the subject like Bezout?s Theorem, the Fundamental Theorem of Projective Geometry, and Zariski?s Main Theorem.
The book contains a detailed treatment of algebraic plane curves with a special emphasis on elliptic curves and their birational classification. The role played by elliptic curves in modern theory of cryptology is illustrated.
A novel feature of the book is a discussion of the state of the art on the Jacobian Problem and its relation to the Epimorphism Theorem. The recently introduced Tame Transformation Method of Cryptosystems, is sketched.
Prerequisities are limited to a knowledge of finite Galois Theory, and of commutative Noetherian rings. All the Commutative Algebra needed is presented in Chapter 1, and could form the basis for a mini course on the subject. The exposition retains classroom flavour. About 300 exercises are included, often with adequate hints
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University of Hyderabad, Hyderabad
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1. Commutative Algebra 1.0 Nakayama Lemma 1.1 Hilbert Basis Theorem 1.2 Localisation 1.3 Graded Rings 1.4 Dimension Theory 1.5 Integral Extensions 1.6 Regular and Normal Rings 1.7 Schmidt and Luroth Theorems 1.8 Elimination Theory 1.9 Exercises
2. Affine Varieties 2.1 Affine Algebraic Sets 2.2 Regular Functions 2.3 Irreducible Algebraic Sets 2.4 Affine Varieties 2.5 Complete Intersections 2.6 Examples 2.7 Morphisms 2.8 Products 2.9 Exercises
3. Projective Varieties 3.1 Terminology 3.2 Projective Algebraic Sets 3.3 Homogenisation/Dehomogenisation 3.4 Projective Closures 3.5 Morphisms 3.6 Products 3.7 Complete Varieties 3.8 Exercises
4. Non-Singular Varieties 4.1 Tangent Spaces 4.2 Jacobian Criterion 4.3 Etale Morphisms 4.4 Tangent Cones 4.5 Blowing-Up a Point 4.6 Exercises
5. Plane Curves 5.1 Generalities 5.2 Rational Curves 5.3 Multiple Points 5.4 Bezout's Theorem 5.5 Applications of Bezout's Theorem 5.6 Elliptic Curves and Complex Tori 5.7 Exercises
6. Zariski's Main Theorem 6.1 Fibre Dimension of Morphisms 6.2 Finite Morphisms 6.3 Normal Varieties 6.4 Normalisation of Varieties 6.5 Zariki's Main Theorem 6.6 Jacobian Problem 6.7 Epimorphism Theorem 6.8 Exercises
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