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1 - Three lectures on Cox rings

from PART ONE - LECTURE NOTES

Published online by Cambridge University Press:  05 May 2013

By
J. Hausen
Edited by
Alexei N. Skorobogatov
J. Hausen
Affiliation:
Universität Tübingen
Alexei N. Skorobogatov
Affiliation:
Imperial College London
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Torsors, Étale Homotopy and Applications to Rational Points , pp. 3 - 60
Publisher: Cambridge University Press
Print publication year: 2013

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References

[1] A., A'Campo-Neuen, J., Hausen: Toric prevarieties and subtorus actions. Geom. Dedicata 87 (2001), no. 1-3, 35–64.Google Scholar
[2] D.F., Anderson: Graded Krull domains. Comm. Algebra 7 (1979), no. 1, 79–106.Google Scholar
[3] M., Artebani, J., Hausen, A., Laface: On Cox rings of K3 surfaces. Compos. Math. 146 (2010), no. 4, 964–998.Google Scholar
[4] I., Arzhantsev, U., Derenthal, J., Hausen, A., Laface: Cox rings. arXiv:1003.4229.
[5] I.V., Arzhantsev, J., Hausen: Geometric Invariant Theory via Cox rings. J. Pure Appl. Algebra 213 (2009), no. 1, 154–172.Google Scholar
[6] M., Audin: The topology of torus actions on symplectic manifolds. Prog. Math., 93. Birkhäuser Verlag, Basel, 1991.
[7] V.V., Batyrev: Quantum cohomology rings of toric manifolds. In: Journées de Géométrie Algébrique d'Orsay, Astérisque 218, 9–34 (1993).Google Scholar
[8] A., Borel: Linear algebraic groups. Second edition. Graduate Texts in Mathematics 126, Springer-Verlag, New York, 1991.
[9] F., Berchtold, J., Hausen: GIT-equivalence beyond the ample cone. Michigan Math. J. 54 (2006), no. 3, 483–515.Google Scholar
[10] F., Berchtold, J., Hausen: Cox rings and combinatorics. Trans. Amer. Math. Soc. 359 (2007), no. 3, 1205–1252.Google Scholar
[11] C., Birkar, P., Cascini, C., Hacon, J., McKernan: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405–468.Google Scholar
[12] J.L., Colliot-Thélène, J.-J., Sansuc: Torseurs sous des groupes de type multiplicatif; applications à l'étude des points rationnels de certaines variétés algébriques. C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 18, Aii, A1113-A1116.Google Scholar
[13] D.A., Cox: The homogeneous coordinate ring of a toric variety. J. Alg. Geom. 4 (1995), no. 1, 17–50.Google Scholar
[14] D.A., Cox, J.B., Little, H.K., Schenck: Toric Varieties. Graduate Studies in Mathematics 124, American Mathematical Society (2011).
[15] V.I., Danilov: The geometry of toric varieties. Uspekhi Mat. Nauk 33 (1978), no. 2 (200), 85–134 (Russian); English transl: Russian Math. Surveys 33 (1978), no. 2, 97-154.Google Scholar
[16] U., Derenthal: Singular Del Pezzo surfaces whose universal torsors are hypersurfaces. arXiv:math/0604194, 2006.
[17] I., Dolgachev: Weighted projective varieties. Group actions and vector fields (Vancouver, B.C., 1981), 34–71, Lecture Notes in Math., 956, Springer, Berlin, 1982.
[18] W., Fulton: Introduction to Toric Varieties. Princeton Univ. Press, Princeton, 1993.
[19] J., Hausen: Cox rings and combinatorics II. Mosc. Math. J. 8 (2008), no. 4, 711–757.Google Scholar
[20] J., Hausen, E., Herppich: Factorially graded rings. This volume.
[21] J., Hausen, E., Herppich, H., Süß;: Multigraded factorial rings and Fano varieties with torus action. Documenta Math. 16 (2011) 71–109.Google Scholar
[22] J., Hausen, H., Süß: The Cox ring of an algebraic variety with torus action. Advances Math. 225 (2010), 977–1012.Google Scholar
[23] B., Hassett, Yu., Tschinkel: Universal torsors and Cox rings. In: Arithmetic of higher-dimensional algebraic varieties, Progr. Math., 226, Birkhäuser Boston, Boston, MA, 149–173 (2004).
[24] E., Herppich: Rational K*-surfaces. In preparation.
[25] Y., Hu, S., Keel: Mori dream spaces and GIT. Michigan Math. J. 48 (2000) 331–348.Google Scholar
[26] A.R., Iano-Fletcher: Working with weighted complete intersections. Explicit birational geometry of 3-folds, 101-173, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000.
[27] M.-N., Ishida: Graded factorial rings of dimension 3 of a restricted type. J. Math. Kyoto Univ. 17 (1977), no. 3, 441–456.Google Scholar
[28] H., Kraft: Geometrische Methoden in der Invariantentheorie. Vieweg Verlag, Braunschweig, 1984.
[29] J., McKernan: Mori dream spaces. Jpn. J. Math. 5 (2010), no. 1, 127–151.Google Scholar
[30] S., Mori: Graded factorial domains. Japan J. Math. 3 (1977), no. 2, 223–238.Google Scholar
[31] D., Mumford, J., Fogarty, F., Kirwan: Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 34. Springer-Verlag, Berlin, 1994.
[32] I.M., Musson: Differential operators on toric varieties. J. Pure Appl. Algebra 95 (1994), no. 3, 303–315.Google Scholar
[33] T., Oda: Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1988.
[34] T., Oda, H.S., Park: Linear Gale transforms and Gelfand-Kapranov-Zelevinskij decompositions. Tohoku Math. J. (2) 43 (1991), no. 3, 375–399.Google Scholar
[35] A.L., Onishchik, E.B., Vinberg: Lie Groups and Algebraic Groups. Springer-Verlag, Berlin Heidelberg, 1990.
[36] J.C., Ottem: On the Cox ring of ℙ2 blown up in points on a line. Math. Scand. 109 (2011), 22–30.Google Scholar
[37] P., Orlik, P., Wagreich: Algebraic surfaces with fc*-action. Acta Math. 138 (1977), no. 1-2, 43–81.Google Scholar
[38] A.N., Skorobogatov: Torsors and rational points. Cambridge Tracts in Math. 144, Cambridge University Press, 2001.
[39] T.A., Springer: Linear algebraic groups. Reprint of the 1998 second edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2009.
[40] H., Sumihiro: Equivariant completion. J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
[41] H., Süß: Canonical divisors on T-varieties. Preprint, arXiv:0811.0626.
[42] J., Tevelev: Compactifications of subvarieties of tori. Amer. J. Math. 129 (2007), 1084–1104.Google Scholar
[43] B., Totaro: The cone conjecture for Calabi-Yau pairs in dimension 2. Duke Math. J. 154 (2010), no. 2, 241–263.Google Scholar
[44] J., Wlodarczyk: Embeddings in toric varieties and prevarieties. J. Alg. Geom. 2 (1993), no. 4, 705–726.Google Scholar

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