Basic Algebraic Geometry, Book 1

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Springer-Verlag, 1994 - Mathematics - 328 pages
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Volume 1: Varieties in Projective Space is an introduction to the theory of algebraic varieties in projective space. Paper. DLC: Geometry - Algebraic.

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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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