Preface
This volume originated from a special session on singularities in algebraic and
analytic geometry, which took place in 1999 at the national American Mathematical
Society meeting in San Antonio, Texas. The year 1999 also marks the hundredth
anniversary of the birth of the great algebraic geometer, Oscar Zariski. It seems
especially fitting at this time to publish a collection of papers dedicated to the
development of his ideas.
Zariski's influence on the study of singularities was evident both in the topics
discussed and in the mathematical heritage of the presenters. Several talks related
to fundamental work on resolution of singularities done by Zariski in the 1930's
and 40's. Shreeram Abhyankar spoke about the history of the subject from a
personal perspective, describing his voyage from India to the United States, how
he met Zariski and became his student, and his subsequent work on resolution of
singularities in characteristic
p.
Three of the contributors to this volume, Ban,
Melles, and Roberts, were students of Zariski's students, and a fourth, McEwan, is
a mathematical great-grandson.
At the time Zariski himself was a student in Rome, that city was the world's
leading center of algebraic geometry. But a few years later, while at Johns Hopkins
University in Baltimore, Zariski realized that the methods of the Italian school were
lacking in rigor, and that to reach a deeper understanding of algebraic geometry he
would need to use methods of commutative algebra to rebuild the entire foundation
of the subject. Commutative algebra is the setting for three of the papers in this
volume, those of Roberts, Vitulli, and Wiegand. Vitulli recounts how the concepts
of weak normalization and weak subintegral closure arose from problems in clas-
sification of algebraic varieties, and proves a conjecture relating weak subintegral
closures of ideals to Rees valuations. In a closely related paper, Roberts describes
two equivalent ways to construct a universal weakly subintegral extension of a ring.
Wiegand's paper on direct-sum decompositions of finitely generated modules is
sprinkled with examples which illustrate how concrete algebraic methods can be.
Zariski 's approach to the problem of resolution of singularities was much more
algebraic than those of his predecessors. He gave algebraic proofs of resolution of
singularities of surfaces and of local uniformization in all dimensions, using valu-
ations of function fields. He introduced the modern notion of the blow-up of an
ideal and recognized the importance of blow-ups of smooth centers. His work made
possible later major developments, including the proof by his student Hironaka of
the existence of resolution of singularities in characteristic zero in all dimensions.
Cutkosky's article offers a modern view of valuations, highlighting the role valua-
tion theory has played in algebraic geometry, outlining Zariski's proof of resolution
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