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