The common solutions of a finite number of polynomial equations in a finite number of variables constitute an algebraic variety. The degrees of freedom of a moving point on the variety is the dimension of the variety. A one-dimensional variety is a curve and a two-dimensional variety is a surface. A three-dimensional variety may be called a solid. Most points of a variety are simple points. Singularities are special points, or points of multiplicity greater than one. Points of multiplicity two are double points, points of multiplicity three are triple points, and so on. A nodal point of a curve is a double point where the curve crosses itself, such as the alpha curve. Acusp is a double point where the curve has a beak. The vertex of a cone provides an example of a surface singularity. A reversible change of variables gives a birational transformation of a variety. Singularities: of a variety maybe resolved by birational transformations.
The present book contains the geometric part of the proof of solid desingularization in characteristic p Z ,3,5 which I obtained in 1965; the algorithmic part is contained in my four previous articles 5 to 9; the book does contain an alternative simple version of the algorithm for characteristic zero; half of the book can also be used as an introduction to birational algebraic geometry.