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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