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Introduction to Singularities (Hardcover, 2nd ed. 2018)
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Introduction to Singularities (Hardcover, 2nd ed. 2018)
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This book is an introduction to singularities for graduate students
and researchers. Algebraic geometry is said to have 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. First, mostly non-singular varieties were
studied. 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. 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-dimensional
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. In the second edition, brief descriptions about recent
remarkable developments of the researches are added as the last
chapter.
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