Algebraic geometry for beginners (Q5941852)

This book provides a self contained introduction to some basic and elementary concepts of algebraic geometry using the classical language. It covers enough material to contain a complete proof of Zariski's main theorem and a discussion of the Jacobian problem (a topic which is for the first time included in a textbook). The pre-requisites are kept to a minimum: A rudimentary familiarity with linear algebra, rings and modules is assumed. Moreover, everything stated is proved as quickly as possible and the main text does not depend on the exercises (given at the end of each chapter). The first part of the book (chapter 1) is a summary (with complete proofs) of the facts needed from commutative algebra. The middle part (chapters 2, 3 and 4) is devoted to setting-up the language: affine and projective varieties, morphisms, rational maps, tangent spaces and cones, singular/non-singular points, étale morphisms, blowing-up a point etc. It contains many examples (including monomial curves, determinantal varieties, group varieties) and a discussion of some problems concerning (set-theoretic) complete intersections. Chapter 5 is devoted to the study of plane curves. It spans several aspects: rational curves, multiple points and the bound for the number of double points in terms of the degree, Bézout's theorem (proved via the classical method of elimination theory) and its standard applications, elliptic curves via the Weierstrass model and a brief account of their applications to cryptography. The final chapter 6 contains a proof of Zariski's main theorem (including basic facts about (quasi-)finite morphisms, semi-continuity theorems, normalization) and a discussion of the Jacobian problem and of the epimorphism theorem of Abhyankar and Moh. The recently introduced tame transformation method of cryptosystems is sketched. As a conclusion: this is a student friendly introduction to algebraic geometry. The author's intention was to fill a gap in the existing literature on the subject consisting of text books which might be considered as too demanding by most of the students.