The notion of singularity is basic to mathematics. In elementary algebra
singularity appears as a multiple root of a polynomial. In geometry a point
in a space is non-singular if it has a tangent space whose dimension is the
same as that of the space. Both notions of singularity can be detected
through the vanishing of derivitives.
Over an algebraically closed field, a variety is non-singular at a point
if there exists a tangent space at the point which has the same dimension
as the variety. More generally, a variety is non-singular at a point if its
local ring is a regular local ring. A fundamental problem is to remove a
singularity by simple algebraic mappings. That is, can a given variety be
desingularized by a proper, birational morphism from a non-singular variety?
This is always possible in all dimensions, over fields of characteristic zero.
We give a complete proof of this in Chapter 6.
We also treat positive characteristic, developing the basic tools needed
for this study, and giving a proof of resolution of surface singularities in
positive characteristic in Chapter 7.
In Section 2.5 we discuss important open problems, such as resolution
of singularities in positive characteristic and local monomialization of mor-
Chapter 8 gives a classification of valuations in algebraic function fields
of surfaces, and a modernization of Zariski's original proof of local uni-
formization for surfaces in characteristic zero.
This book has evolved out of lectures given at the University of Mis-
souri and at the Chennai Mathematics Institute, in Chennai, (also known
as Madras), India. It can be used as part of a one year introductory sequence