X
PREFACE
of surface singularities, and concluding with recent applications to the structure of
morphisms.
Algorithmic approaches to resolution of singularities are featured in the papers
of Ban and McEwan, and of Melles and Milman. Ban and McEwan prove simul-
taneous resolution of equisingular surface singularities using the algorithm given
by Bierstone and Milman in their constructive and canonical proof of Hironaka's
theorem. Melles and Milman describe combinatorial blow-ups and outline their use
in resolving the singularities of locally toric varieties.
The papers of Nemethi and Szilard, on resolution graphs of surface singularities,
also employ an algorithmic approach. Their examples and classification theorems
illustrate the richness of the theory of surface singularities.
The theory of curves is another beautiful and fertile area. Abhyankar and
Assi discuss recent work on Jacobians of meromorphic curves. McEwan examines
spectral numbers of curve singularities, using Puiseux pairs.
Hypersurfaces form yet another class of spaces which are amenable to spe-
cialized techniques and provide a valuable testing ground for conjectures. Michler
uses Grobner basis calculations of Hochschild homology to investigate hypersurfaces
with isolated singularities.
While Zariski was rebuilding the foundations of algebraic geometry, Severi and
Todd, and then Chern, introduced characteristic classes as a way to study structures
on complex algebraic varieties. Brasselet gives a brief history of the theory of
characteristic classes of singular spaces and discusses some conjectures and open
problems.
The editors are indebted to the contributors to the special session and to this
volume, for their generosity and enthusiasm; to the referees, for their helpful sug-
gestions, often under short deadlines; to the American Mathematical Society for its
support of the special session and these proceedings; and to Christine M. Thivierge,
acquisitions assistant at the AMS, for her editorial advice, patience, and encour-
agement.
In acknowledgment of his great contributions to algebraic geometry, this volume
is dedicated to Oscar Zariski.
Caroline Grant Melles
Ruth Michler
May 2000
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