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These notes are an introduction to the theory of algebraic varieties emphasizing the similarities to the theory of manifolds. In contrast to most such accounts they study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory. Before learning scheme theory everyone should understand algebraic varieties over algebraically closed fields: first the geometric intuition and then the abstractions. Algebraic varieties over algebraically closed fields are the reduced geometric fibres of morphisms of schemes.
Prerequisites
The reader is assumed to be familiar with the basic objects of algebra, namely, rings, modules, fields, and so on.
Introduction.
Preliminaries from commutative algebra.
Algebraic Sets.
Affine Algebraic Varieties.
Local Study.
Algebraic Varieties.
Projective Varieties.
Complete Varieties.
Normal Varieties; (Quasi-)finite maps; Zariski’s Main Theorem.
Regular Maps and Their Fibres.
Solutions to the exercises.
Index