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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.
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.
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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