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In this book we study Hilbert schemes of zero-dimensional
subschemes of smooth varieties and several related parameter
varieties of interest in enumerative geometry. The main aim here is
to describe their cohomology and Chow rings. Some enumerative
applications are also given. The Weil conjectures are used to
compute the Betti numbers of many of the varieties considered, thus
also illustrating how this powerful tool can be applied. The book
is essentially self-contained, assuming only a basic knowledge of
algebraic geometry; it is intended both for graduate students and
research mathematicians interested in Hilbert schemes, enumertive
geometry and moduli spaces.
Alexander Grothendieck's concepts turned out to be astoundingly
powerful and productive, truly revolutionizing algebraic geometry.
He sketched his new theories in talks given at the Seminaire
Bourbaki between 1957 and 1962. He then collected these lectures in
a series of articles in Fondements de la geometrie algebrique
(commonly known as FGA).
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