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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - I. V. Arzhantsev
TI  - On the Factoriality of Cox rings
JO  - Matematičeskie zametki
PY  - 2009
SP  - 643
EP  - 651
VL  - 85
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a0/
LA  - ru
ID  - MZM_2009_85_5_a0
ER  - 
%0 Journal Article
%A I. V. Arzhantsev
%T On the Factoriality of Cox rings
%J Matematičeskie zametki
%D 2009
%P 643-651
%V 85
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2009_85_5_a0/
%G ru
%F MZM_2009_85_5_a0
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/

[1] D. A. Cox, “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom., 4:1 (1995), 17–50 | MR | Zbl

[2] F. Berchtold, J. Hausen, “Homogeneous coordinates for algebraic varieties”, J. Algebra, 266:2 (2003), 636–670 | DOI | MR | Zbl

[3] E. J. Elizondo, K. Kurano, K. Watanabe, “The total coordinate ring of a normal projective variety”, J. Algebra, 276:2 (2004), 625–637 | DOI | MR | Zbl

[4] J. Hausen, Cox rings and combinatorics. II, arXiv: math.AG/0801.3995

[5] D. A. Timashev, Homogeneous spaces and equivariant embeddings, arXiv: math.AG/0602228

[6] F. Knop, H. Kraft, D. Luna, Th. Vust, “Local properties of algebraic group actions”, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., 13, Birkhäuser, Basel, 1989, 63–75 | MR | Zbl

[7] F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, Lecture Notes in Math., 1673, Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

[8] P. Samuel, Lectures on unique factorization domains, Notes by M. Pavman Murthy. Tata Inst. Fund. Res. Lectures on Math., 30, Tata Institute of Fundamental Research, Bombay, 1964 | MR | Zbl

[9] V. L. Popov, “Gruppy Pikara odnorodnykh prostranstv lineinykh algebraicheskikh grupp i odnomernye odnorodnye vektornye rassloeniya”, Izv. AN SSSR. Ser. matem., 38:2 (1974), 294–322 | MR | Zbl

[10] I. V. Arzhantsev, J. Hausen, “On embeddings of homogeneous spaces with small boundary”, J. Algebra, 304:2 (2006), 950–988 | DOI | MR | Zbl

[11] M. Brion, “The total coordinate ring of a wonderful variety”, J. Algebra, 313:1 (2007), 61–99 | DOI | MR | Zbl