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On the factoriality of Cox rings

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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.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    I. V. Arzhantsev

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  1. I. V. Arzhantsev
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Correspondence to I. V. Arzhantsev.

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Original Russian Text © I. V. Arzhantsev, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 5, pp. 643–651.

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Arzhantsev, I.V. On the factoriality of Cox rings. Math Notes 85, 623–629 (2009). https://doi.org/10.1134/S0001434609050022

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  • DOI: https://doi.org/10.1134/S0001434609050022

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