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Mathematical Notes

, Volume 85, Issue 5–6, pp 623–629 | Cite as

On the factoriality of Cox rings

  • I. V. Arzhantsev
Article

Abstract

The generalized Cox construction associates with an algebraic variety a remarkable invariant—its total coordinate ring, or Cox ring. In this note, we give a new proof of the factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on the notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost.

Key words

total coordinate ring Cox ring algebraic variety factorial ring graded factoriality divisor class group torsion Weil divisor Cartier divisor 

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Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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