This new edition describes the geometric part of the authors 1965 proof ofdesingularization of algebraic surfaces and solids in nonzero characteristic.The book ... » 

This new edition describes the geometric part of the authors 1965 proof ofdesingularization of algebraic surfaces and solids in nonzero characteristic.The book also provides a selfcontained introduction to birational algebraicgeometry based only on basic commutative algebra. In addition it gives ashort proof of analytic desingularization in characteristic zero for anydimension found in 1996 and based on a new avatar of an algorithmic trickemployed in the original edition of the book. This new edition will inspirefurther progress in resolution of singularities of algebraic and arithmeticalvarieties which will be valuable for applications to algebraic geometry andnumber theory. The book can be used for a second year graduate course. Thereference list has been updated. «