Basic Algebraic Geometry, Book 1

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Springer-Verlag, 1994 - Mathematics - 328 pages
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The section on singularities of a map (starting in page 139) makes interesting reading. Particularly interesting is the mention of examples of irreducible algebraic varieties V that become reducible over the algebraic closure of their function field. A nice example of this is given. Two nice theorem's in this section are: The First Bertini Theorem gives criteria that guarantees the prevention of the mentioned anomaly in the case the field of definition of V is of characteristic zero. The second Bertini's Theorem guarantees that a dense map f from a nonsingular variety X to a variety Y will have an open set U in its image with the property that all fibers over individual points of U are non-singular.  

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About the author (1994)

Igor Rostislavovich Shafarevich was born in Zhitomir, Ukraine on June 3, 1923. He graduated from Moscow State University with a specialty in astronomy. He taught at Moscow State University for more than 30 years. He was an internationally renowned mathematician who played a central role in the anti-Soviet dissident movement during the Cold War. His textbooks on algebraic geometry were translated into English and regarded as classics in the field. He also wrote The Socialist Phenomenon and contributed essays to From Under the Rubble. He died on February 19, 2017 at the age of 93.

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