The aim of the book is to provide a concise introduction to algebraic geometry and algebraic moduli theory. On the way to this goal, the author explains some of the fundamental contributions of Caley, Hilbert, Nagata, Grothendieck and Mumford, keeping proofs as elementary as possible or avoiding them completely. The author works in the category of algebraic varieties instead of schemes and sheaves (regarded as functors) and uses quotients of affine algebraic varieties by general linear group GL instead of geometric invariant quotients of projective varieties by PGL. In constructing the moduli of vector bundles on an algebraic curve, the Grothendieck Quot scheme is replaced by certain explicit affine variety consisting of matrices with polynomial entries. Important analytic viewpoints represented by Hodge and Kodaira-Spencer theories are not treated in the book.

Reviewer:

pso