Chapter 1 Introduction An algebraic variety X is defined locally by the vanishing of a system of polynomial equations fa G K[xi,..., xn], / l = • • • = / m = 0 . If K is algebraically closed, points of X in this chart are a = (ai,... , a n ) G A^ which satisfy this system. The tangent space Ta(X) at a point a G X is the linear subspace of A^ defined by the system of linear equations L\ = • • • = Lm = 0, where Li is defined by We have that dim Ta(X) dim X, and X is non-singular at the point a if dim Ta(X) — dim X. The locus of points in X which are singular is a proper closed subset of X. The fundamental problem of resolution of singularities is to perform simple algebraic transformations of X so that the transform Y of X is non- singular everywhere. To be precise, we seek a resolution of singularities of X] that is, a proper birational morphism $ : Y — X such that Y is non-singular. The problem of resolution when K has characteristic zero has been stud- ied for some time. In fact we will see (Chapters 2 and 3) that the method of Newton for determining the analytical branches of a plane curve singularity extends to give a proof of resolution for algebraic curves. The first algebraic proof of resolution of surface singularities is due to Zariski [86]. We give 1 http://dx.doi.org/10.1090/gsm/063/01
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