Resolution of Singularities
An algebraic variety X over an algebraically closed field K need not be a smooth manifold. Its set of singular points is a proper closed subset of X. The famous result of Hironaka from 60’s says that there is a resolution of singularities of X, if K is a field of characteristic zero. It means that there exists a non-singular variety Y and a projective morphism π of Y on X, which is an isomorphism away from the singular locus of X. The existence of a resolution of singularities has a number of important applications in other fields of mathematics and mathematical physics. The presented book contains a discussion of resolution of singularities in various cases. The first six chapters of the book contain the proof of the Hironaka desingularization theorem based on canonical resolutions. The last three chapters cover additional topics (including resolutions of singularities for surfaces in positive characteristic and resolutions of surface singularities through local uniformization of valuations). The reader is assumed to know basic facts from algebraic geometry (schemes) and commutative algebra.