Smith K.E., Kahanpää L., Kekäläinen P., Traves W. An invitation to algebraic geometry

Springer, 2000. — 173 p. — (Universitext). — ISBN: 0387989803.The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today. It is intended for the working or the aspiring mathematician who is unfamiliar with algebraic geometry but wishes to gain an appreciation of its foundations and its goals with a minimum of prerequisites. Few algebraic prerequisites are presumed beyond a basic course in linear algebra.Affine Algebraic Varieties
Algebraic Foundations
Projective Varieties
Quasi-Projective Varieties
Classical Constructions
Smoothness
Birational Geometry
Maps to Projective Space