The exposition of the subject of algebraic geometry in this book does not pass through sheaf theory, cohomology, derived functors, categories or abstract commutative algebras but it is rather focused on specific examples, together with a part of the basic formalism that is most useful for computations. In particular, Gröbner bases are introduced quite early and for almost every technique there is both an algorithmic and a computational approach. All core techniques of algebraic geometry are developed systematically from scratch, with necessary commutative algebra integrated to geometry. Classical topics (like resultants and elimination theory) are discussed in parallel with affine varieties, morphisms and rational maps. Important examples of projective varieties (like Grassmannians, Veronese and Segre varieties) are emphasised, along with the matrix and exterior algebras needed to write down their defining equations.

Reviewer:

pso