Resolution of curve and surface singularities in characteristic zero.
The monograph presents a modern theory of resolution of isolated singularities of algebraic curves and surfaces over algebraically closed fields of characteristic zero. The authors develop appropriate algebraic techniques, the valuation theory of rings and fields, ideals of one- and two-dimensional rings, differential modules and ramifications, algebras of formal and convergent power series, and then treat the resolution of singularities of curves and the resolution of isolated normal surface singularities. The emphasis is made on a controlled resolution process, presented by a finite sequence of birational transformations. For curve singularities, it results in successive blowing-ups of the curve embedded into a projective space. In turn the resolution of surface singularities can be performed in two ways. One algorithm follows the Jung idea. It is based on projecting the surface onto a plane with a normal crossing discriminant locus and then parameterizing a complete neighborhood of the singularity by conjugate Puiseux power series. Another algorithm is that of Zariski, and it amounts in a sequence of blowing-ups and normalizations, governed by valuations which can be uniformized. The exposition is self-contained and is supplied by an appendix, covering some classical algebraic geometry and commutative algebra.
scientific article; zbMATH DE number 2146319