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Books > Science & Mathematics > Mathematics > Algebra
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Introduction to Singularities (Paperback, Softcover reprint of the original 1st ed. 2014)
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Introduction to Singularities (Paperback, Softcover reprint of the original 1st ed. 2014)
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This book is an introduction to singularities for graduate students
and researchers. It is said that algebraic geometry originated in
the seventeenth century with the famous work Discours de la methode
pour bien conduire sa raison, et chercher la verite dans les
sciences by Descartes. In that book he introduced coordinates to
the study of geometry. After its publication, research on algebraic
varieties developed steadily. Many beautiful results emerged in
mathematicians' works. Most of them were about non-singular
varieties. Singularities were considered "bad" objects that
interfered with knowledge of the structure of an algebraic variety.
In the past three decades, however, it has become clear that
singularities are necessary for us to have a good description of
the framework of varieties. For example, it is impossible to
formulate minimal model theory for higher-dimensional cases without
singularities. Another example is that the moduli spaces of
varieties have natural compactification, the boundaries of which
correspond to singular varieties. A remarkable fact is that the
study of singularities is developing and people are beginning to
see that singularities are interesting and can be handled by human
beings. This book is a handy introduction to singularities for
anyone interested in singularities. The focus is on an isolated
singularity in an algebraic variety. After preparation of
varieties, sheaves, and homological algebra, some known results
about 2-dim ensional isolated singularities are introduced. Then a
classification of higher-dimensional isolated singularities is
shown according to plurigenera and the behavior of singularities
under a deformation is studied.
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