Titlebook: An Invitation to Algebraic Geometry; Karen E. Smith,Lauri Kahanpää,William Traves Textbook 2000 Springer Science+Business Media New York 2

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Otto Körner Dr. med., Dr. phil. h. c.Much of the power and rigor of algebraic geometry comes from the fact that geometric questions can be translated into purely algebraic problems.

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,Die Gruppe der Schlangen (öϕιες),Affine space A. has a natural compactification, the projective space ℙ., obtained by adding an infinitely distant point in every direction. The goal of this chapter is to introduce projective space and projective varieties and to interpret them as natural compactifications of affine varieties.

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,Störungen des visuellen Erkennens,Veronese maps provide an important example of morphisms of quasi-projective varieties. A Veronese map embeds a projective space ℙ. as a subvariety of some higher-dimensional projective space in a nontrivial way.

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Projective Varieties,Affine space A. has a natural compactification, the projective space ℙ., obtained by adding an infinitely distant point in every direction. The goal of this chapter is to introduce projective space and projective varieties and to interpret them as natural compactifications of affine varieties.

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Classical Constructions,Veronese maps provide an important example of morphisms of quasi-projective varieties. A Veronese map embeds a projective space ℙ. as a subvariety of some higher-dimensional projective space in a nontrivial way.

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Birational Geometry,In 1964, Heisuke Hironaka proved a fundamental theorem: Every quasi-projective variety can be ., or equivalently, every variety is “birationally equivalent” to a smooth projective variety. Before we can state this theorem, we need to introduce some new ideas.