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Homogeneous coordinates for algebraic varieties

✍ Scribed by Florian Berchtold; Jürgen Hausen

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
301 KB
Volume
266
Category
Article
ISSN
0021-8693

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✦ Synopsis

We associate to every divisorial (e.g., smooth) variety X with only constant invertible global functions and finitely generated Picard group a Pic(X)-graded homogeneous coordinate ring. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of Cox and Kajiwara for smooth and divisorial toric varieties. We show that the homogeneous coordinate ring defines in fact a fully faithful functor. For normal complex varieties X with only constant global functions, we even obtain an equivalence of categories. Finally, the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated Picard group turns out to be a Krull ring admitting unique factorization.

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