This book contains a collection of reprints of important classical papers written by David Mumford. The book is divided into three parts, each of them coming with commentaries by leading experts in the field. The first part contains the Mumford study of various aspects of the moduli space of algebraic curves. He was the first to formulate a purely algebraic approach (based on geometric invariant theory and valid in all characteristics) to the description of the moduli space of Riemann surfaces as a quasi-projective variety. It also applies to Chow rings, tautological classes, enumerative geometry and compactification of the moduli spaces of algebraic curves by stable (nodal) algebraic curves.

The second part is devoted to Mumford’s work on (finite) theta functions and equations of Abelian varieties, families of Abelian varieties and their degeneracies, theta characteristics, the Horrocks-Mumford bundle, Prym varieties and compactifications of bounded symmetric domains. The third series of Mumford's articles focuses on the classification of surfaces and other special varieties. It involves vanishing theorems and pathologies in positive characteristic for complex projective varieties and a classification theory of surfaces in positive characteristics. The Iitaka and the Mori minimal model program as well as many explicit (non-)existence results are included in a series of articles on pathologies in algebraic geometry.

Reviewer:

pso