Geodesic
On the Factoriality of Cox rings
Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 643-651

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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.
Keywords: total coordinate ring, Cox ring, algebraic variety, factorial ring, graded factoriality, divisor class group, Weil divisor
Mots-clés : torsion, Cartier divisor. 
@article{MZM_2009_85_5_a0,
     author = {I. V. Arzhantsev},
     title = {On the {Factoriality} of {Cox} rings},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--651},
     publisher = {mathdoc},
     volume = {85},
     number = {5},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a0/}
}
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I. V. Arzhantsev. On the Factoriality of Cox rings. Matematičeskie zametki, Tome 85 (2009) no. 5, pp. 643-651. http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a0/