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Cox Rings
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Book description

Cox rings are significant global invariants of algebraic varieties, naturally generalizing homogeneous coordinate rings of projective spaces. This book provides a largely self-contained introduction to Cox rings, with a particular focus on concrete aspects of the theory. Besides the rigorous presentation of the basic concepts, other central topics include the case of finitely generated Cox rings and its relation to toric geometry; various classes of varieties with group actions; the surface case; and applications in arithmetic problems, in particular Manin's conjecture. The introductory chapters require only basic knowledge in algebraic geometry. The more advanced chapters also touch on algebraic groups, surface theory, and arithmetic geometry. Each chapter ends with exercises and problems. These comprise mini-tutorials and examples complementing the text, guided exercises for topics not discussed in the text, and, finally, several open problems of varying difficulty.

Reviews

‘An excellent introduction to the subject, featuring a wide selection of topics, careful exposition, and many examples and exercises.'

David Cox - University of Massachusetts, Amherst

‘This book is a detailed account of virtually every aspect of the general theory of the Cox ring of an algebraic variety. After a thorough introduction it takes the reader on an impressive tour through toric geometry, geometric invariant theory, Mori dream spaces, and universal torsors, culminating with applications to the Manin conjecture on rational points. The many worked examples and exercises make it not just a comprehensive reference, but also an excellent introduction for graduate students.'

Alexei Skorobogatov - Imperial College London

‘This book provides the first comprehensive treatment of Cox rings. Firstly, its broad and complete exposition of the fundamentals of the general theory will be appreciated by both those who want to learn the subject and specialists seeking an ultimate reference on many subtle aspects of the theory. Secondly, it introduces readers to the most important applications that have developed in the past decade and will define the direction of research in the years to come.'

Jarosław Wiśniewski - Institute of Mathematics, University of Warsaw

'Cox rings are very important in modern algebraic and arithmetic geometry. This book, providing a comprehensive introduction to the theory and applications of Cox rings from the basics up to, and including, very complicated technical points and particular problems, aims at a wide readership of more or less everyone working in the areas where Cox rings are used … This book is very useful for everyone working with Cox rings, and especially useful for postgraduate students learning the subject.'

Alexandr V. Pukhlikov Source: Mathematical Reviews

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Contents

References
[1] A., A'Campo-Neuen. Note on a counterexample to Hilbert's fourteenth problem given by P. Roberts. Indag. Math. (N.S.), 5(3):253–257, 1994.
[2] A., A'Campo-Neuen and J., Hausen. Examples and counterexamples for existence of categorical quotients. J. Algebra, 231(1):67–85, 2000.
[3] A., A'Campo-Neuen and J., Hausen. Toric prevarieties and subtorus actions. Geom. Dedicata, 87(1–3):35–64, 2001.
[4] A. G., Aleksandrov and B. Z., Moroz. Complete intersections in relation to a paper of B. J. Birch. Bull. London Math. Soc., 34(2):149–154, 2002.
[5] V., Alexeev and V. V., Nikulin. Del Pezzo and K3 surfaces. MSJ Memoirs, Vol. 15. Mathematical Society of Japan, Tokyo, 2006.
[6] D. F., Anderson. Graded Krull domains. Comm. Algebra, 7(1):79–106, 1979.
[7] E., Arbarello, M., Cornalba, P. A., Griffiths, and J., Harris. Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, Vol. 267. Springer-Verlag, New York, 1985.
[8] M., Artebani, A., Garbagnati, and A., Laface. Cox rings of extremal rational elliptic surfaces. arXiv:1302.4361, 2013.
[9] M., Artebani, J., Hausen, and A., Laface. On Cox rings of K3 surfaces. Compos. Math., 146(4):964–998, 2010.
[10] M., Artebani and A., Laface. Cox rings of surfaces and the anticanonical Iitaka dimension. Adv. Math., 226(6):5252–5267, 2011.
[11] M., Artebani and A., Laface. Hypersurfaces in Mori dream spaces. J. Algebra, 371:26–37, 2012.
[12] M., Artin. Some numerical criteria for contractability of curves on algebraic surfaces. Amer. J. Math., 84:485–496, 1962.
[13] I. V., Arzhantsev. On the factoriality of Cox rings. Mat. Zametki, 85(5):643–651, 2009.
[14] I. V., Arzhantsev. Projective embeddings with a small boundary for homogeneous spaces. Izv. Ross. Akad. Nauk Ser. Mat., 73(3):5–22, 2009.
[15] I. V., Arzhantsev, D., Celik, and J., Hausen. Factorial algebraic group actions and categorical quotients. J. Algebra, 387:87–98, 2013.
[16] I. V., Arzhantsev and S. A., Gaĭfullin. Cox rings, semigroups, and automorphisms of affine varieties. Mat. Sb., 201(1):3–24, 2010.
[17] I. V., Arzhantsev and S. A., Gaĭfullin. Homogeneous toric varieties. J. Lie Theory, 20(2):283–293, 2010.
[18] I. V., Arzhantsev and J., Hausen. On embeddings of homogeneous spaces with small boundary. J. Algebra, 304(2):950–988, 2006.
[19] I. V., Arzhantsev and J., Hausen. On the multiplication map of a multigraded algebra. Math. Res. Lett., 14(1):129–136, 2007.
[20] I. V., Arzhantsev and J., Hausen. Geometric invariant theory via Cox rings. J. Pure Appl. Algebra, 213(1):154–172, 2009.
[21] I. V., Arzhantsev, J., Hausen, E., Herppich, and A., Liendo. The automorphism group of a variety with torus action of complexity one. Mosc. Math. J., 14(3):429–471, 2014.
[22] M. F., Atiyah and I. G., Macdonald. Introduction to commutative algebra. Addison-Wesley, Reading, MA-London-Don Mills, Ont., 1969.
[23] M., Audin. The topology of torus actions on symplectic manifolds. Progress in Mathematics, Vol. 93. Birkhäuser Verlag, Basel, 1991.
[24] R., Avdeev. An epimorphic subgroup arising from Roberts' counterexample. Indag. Math. (N.S.), 23(1–2):10–18, 2012.
[25] S., Baier and T. D., Browning. Inhomogeneous cubic congruences and rational points on del Pezzo surfaces. J. reine angew. Math., 680:69–151, 2013.
[26] S., Baier and U., Derenthal. Quadratic congruences on average and rational points on cubic surfaces. arXiv:1205.0373, 2012.
[27] H., Bäker. Good quotients of Mori dream spaces. Proc. Amer. Math. Soc., 139(9):3135–3139, 2011.
[28] H., Bäker, J., Hausen, and S., Keicher. On Chow quotients of torus actions. arXiv:1203.3759, 2012.
[29] W. P., Barth, K., Hulek, C. A. M., Peters, and A., Van de Ven. Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 4, 2nd ed. Springer-Verlag, Berlin, 2004.
[30] V. V., Batyrev. Quantum cohomology rings of toric manifolds. Astérisque, (218):9–34, 1993. Journeés de Géométrie Algébrique d'Orsay (Orsay, 1992).
[31] V. V., Batyrev and F., Haddad. On the geometry of SL(2)-equivariant flips. Mosc. Math. J., 8(4):621–646, 846, 2008.
[32] V. V., Batyrev and Yu. I., Manin. Sur le nombre des points rationnels de hauteur borné des variétés algébriques. Math. Ann., 286(1–3):27–43, 1990.
[33] V. V., Batyrev and O. N., Popov. The Cox ring of a del Pezzo surface. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002). Progress in Mathematics, Vol. 226, pp. 85–103. Birkhäuser Boston, Boston, 2004.
[34] V. V., Batyrev and Yu., Tschinkel. Rational points of bounded height on compactifications of anisotropic tori. Int. Math. Res. Not. IMRN, (12):591–635, 1995.
[35] V. V., Batyrev and Yu., Tschinkel. Rational points on some Fano cubic bundles. C. R. Acad. Sci. Paris Sér. I Math., 323(1):41–46, 1996.
[36] V. V., Batyrev and Yu., Tschinkel. Manin's conjecture for toric varieties. J. Algebraic Geom., 7(1):15–53, 1998.
[37] V. V., Batyrev and Yu., Tschinkel. Tamagawa numbers of polarized algebraic varieties. Astérisque, (251):299–340, 1998. Nombre et répartition de points de hauteur bornée (Paris, 1996).
[38] S., Bauer. Die Manin-Vermutung für eine del-Pezzo-Fläche, Master's thesis, Universität München, 2013.
[39] A., Beauville. Complex algebraic surfaces. London Mathematical Society Student Texts, Vol. 34, 2nd ed. Cambridge University Press, Cambridge, 1996.
[40] B., Bechtold. Factorially graded rings and Cox rings. J. Algebra, 369:351–359, 2012.
[41] O., Benoist. Quasi-projectivity of Normal Varieties. Int. Math. Res. Not. IMRN, (17):3878–3885, 2013.
[42] F., Berchtold and J., Hausen. Homogeneous coordinates for algebraic varieties. J. Algebra, 266(2):636–670, 2003.
[43] F., Berchtold and J., Hausen. Bunches of cones in the divisor class group—a new combinatorial language for toric varieties. Int. Math. Res. Not. IMRN, (6):261–302, 2004.
[44] F., Berchtold and J., Hausen. Cox rings and combinatorics. Trans. Amer. Math. Soc., 359(3):1205–1252 (electronic), 2007.
[45] A., Bialynicki-Birula. Finiteness of the number of maximal open subsets with good quotients. Transform. Groups, 3(4):301–319, 1998.
[46] A., Bialynicki-Birula, J. B., Carrell, and W. M., McGovern. Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action. Invariant Theory and Algebraic Transformation Groups, II. Encyclopaedia of Mathematical Sciences, Vol. 131. Springer-Verlag, Berlin, 2002.
[47] A., Bialynicki-Birula and J., Święcicka. Three theorems on existence of good quotients. Math. Ann., 307(1):143–149, 1997.
[48] A., Bialynicki-Birula and J., Święcicka. A recipe for finding open subsets of vector spaces with a good quotient. Colloq. Math., 77(1):97–114, 1998.
[49] F., Bien and A., Borel. Sous-groupes epimorphiques de groupes linéaires algébriques. I. C. R. Acad. Sci. Paris Sér. I Math., 315(6):649–653, 1992.
[50] F., Bien and A., Borel. Sous-groupes epimorphiques de groupes linéaires algébriques. II. C. R. Acad. Sci. Paris Sér. I Math., 315(13):1341–1346, 1992.
[51] F., Bien, A., Borel, and J., Kollár. Rationally connected homogeneous spaces. Invent. Math., 124(1–3):103–127, 1996.
[52] B. J., Birch. Forms in many variables. Proc. Roy. Soc. Ser. A, 265:245–263, 1961/1962.
[53] C., Birkar, P., Cascini, C. D., Hacon, and J., McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 23(2):405–468, 2010.
[54] V., Blomer and J., Brüdern. The density of rational points on a certain threefold. In Contributions in analytic and algebraic number theory. Springer Proceedings in Mathematics, Vol. 9, pp. 1–15. Springer-Verlag, New York, 2012.
[55] V., Blomer, J., Brüdern, and P., Salberger. On a senary cubic form. Proc. Lond. Math. Soc. (3), 108(4):911–964, 2014.
[56] A., Borel. Linear algebraic groups. Graduate Texts in Mathematics, Vol. 126, 2nd ed. Springer-Verlag, New York, 1991.
[57] M., Borovoi. The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer. J. reine angew. Math., 473:181–194, 1996.
[58] N., Bourbaki. Commutative algebra. Chapters 1–7. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998.
[59] D., Bourqui. Fonction zêta des hauteurs des variétés toriques déployées dans le cas fonctionnel. J. reine angew. Math., 562:171–199, 2003.
[60] D., Bourqui. Comptage de courbes sur le plan projectif éclaté en trois points alignés. Ann. Inst. Fourier (Grenoble), 59(5):1847–1895, 2009.
[61] D., Bourqui. Fonction zêta des hauteurs des variétés toriques non déployées. Mem. Amer. Math. Soc., 211(994):viii+151, 2011.
[62] D., Bourqui. La conjecture de Manin géométrique pour une famille de quadriques intrinsèques. Manuscripta Math., 135(1–2):1–41, 2011.
[63] D., Bourqui. Exemples de comptage de courbes sur les surfaces. Math. Ann., 357(4):1291–1327, 2013.
[64] J.-F., Boutot. Singularités rationnelles et quotients par les groupes réductifs. Invent. Math., 88(1):65–68, 1987.
[65] H., Brenner and S., Schröer. Ample families, multihomogeneous spectra, and algebraization of formal schemes. Pacific J. Math., 208(2):209–230, 2003.
[66] R., de la Bretèche. Compter des points d'une variété torique. J. Number Theory, 87(2):315–331, 2001.
[67] R., de la Bretèche. Répartition des points rationnels sur la cubique de Segre. Proc. Lond. Math. Soc. (3), 95(1):69–155, 2007.
[68] R., de la Bretèche and T. D., Browning. On Manin's conjecture for singular del Pezzo surfaces of degree 4. I. Michigan Math. J., 55(1):51–80, 2007.
[69] R., de la Bretèche and T. D., Browning. On Manin's conjecture for singular del Pezzo surfaces of degree four. II. Math. Proc. Cambridge Philos. Soc., 143(3):579–605, 2007.
[70] R., de la Bretèche and T. D., Browning. Manin's conjecture for quartic del Pezzo surfaces with a conic fibration. Duke Math. J., 160(1):1–69, 2011.
[71] R., de la Bretèche and T. D., Browning. Binary forms as sums of two squares and Châtelet surfaces. Israel J. Math., 191(2):973–1012, 2012.
[72] R., de la Bretèche, T. D., Browning, and U., Derenthal. On Manin's conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup. (4), 40(1):1–50, 2007.
[73] R., de la Bretèche, T. D., Browning, and E., Peyre. On Manin's conjecture for a family of Châtelet surfaces. Ann. of Math. (2), 175(1):297–343, 2012.
[74] R., de la Bretèche and E., Fouvry. L'éclaté du plan projectif en quatre points dont deux conjugués. J. reine angew. Math., 576:63–122, 2004.
[75] R., de la Bretèche and G., Tenenbaum. Sur la conjecture de Manin pour certaines surfaces de Châtelet. J. Inst. Math. Jussieu, 12(4):759–819, 2013.
[76] M., Brion. The total coordinate ring of a wonderful variety. J. Algebra, 313(1):61–99, 2007.
[77] M., Brion and S., Kumar. Frobenius splitting methods in geometry and representation theory. Progress in Mathematics, Vol. 231. Birkhäuser Boston, Boston, 2005.
[78] T. D., Browning. Quantitative arithmetic of projective varieties. Progress in Mathematics, Vol. 277. Birkhäuser Verlag, Basel, 2009.
[79] T. D., Browning and U., Derenthal. Manin's conjecture for a cubic surface with D5 singularity. Int. Math. Res. Not. IMRN, (14):2620–2647, 2009.
[80] T. D., Browning and U., Derenthal. Manin's conjecture for a quartic del Pezzo surface with A4 singularity. Ann. Inst. Fourier (Grenoble), 59(3):1231–1265, 2009.
[81] T. D., Browning and D. R., Heath-Brown. Quadratic polynomials represented by norm forms. Geom. Funct. Anal., 22(5):1124–1190, 2012.
[82] T. D., Browning and L., Matthiesen. Norm forms for arbitrary number fields as products of linear polynomials. arXiv:1307.7641, 2013.
[83] T. D., Browning, L., Matthiesen, and A. N., Skorobogatov. Rational points on pencils of conics and quadrics with many degenerate fibres. Ann. of Math. 180(1):381–402, 2014.
[84] J. W., Bruce and C. T. C., Wall. On the classification of cubic surfaces. J. London Math. Soc. (2), 19(2):245–256, 1979.
[85] W., Bruns and J., Gubeladze. Polytopal linear groups. J. Algebra, 218(2):715–737, 1999.
[86] A.-M., Castravet and J., Tevelev. Hilbert's 14th problem and Cox rings. Compos. Math., 142(6):1479–1498, 2006.
[87] D., Celik. A categorical quotient in the category of dense constructible subsets. Colloq. Math., 116(2):147–151, 2009.
[88] A., Chambert-Loir and Yu., Tschinkel. On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math., 148(2):421–452, 2002.
[89] R., Chirivì, P., Littelmann, and A., Maffei. Equations defining symmetric varieties and affine Grassmannians. Int. Math.Res.Not.IMRN, (2):291–347, 2009.
[90] R., Chirivì and A., Maffei. The ring of sections of a complete symmetric variety. J. Algebra, 261(2):310–326, 2003.
[91] J.-L., Colliot-Thélène. Points rationnels sur les fibrations. In Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Society Mathematical Studies, Vol. 12, pp. 171–221. Springer-Verlag, Berlin, 2003.
[92] J.-L., Colliot-Thélène. Lectures on linear algebraic groups, Morning Side Centre, Beijing, http://www.math.u-psud.fr/~colliot, 2007.
[93] J.-L., Colliot-Thélène, D., Harari, and A. N., Skorobogatov. Valeurs d'un polynôme a une variable représentées par une norme. In Number theory and algebraic geometry. London Mathematical Society Lecture Note Series, Vol. 303, pp. 69–89. Cambridge University Press, Cambridge, 2003.
[94] J.-L., Colliot-Thélène, A., Pál, and A. N., Skorobogatov. Pathologies of the Brauer-Manin obstruction. arXiv:1310.5055, 2013.
[95] J.-L., Colliot-Thélène and P., Salberger. Arithmetic on some singular cubic hyper-surfaces. Proc. London Math. Soc. (3), 58(3):519–549, 1989.
[96] J.-L., Colliot-Thélène and J.-J., Sansuc. Torseurs sous des groupes de type multiplicatif; applications à l'étude des points rationnels de certaines variétés algebriques. C. R. Acad. Sci. Paris Sér. A-B, 282(18):Aii, A1113–A1116, 1976.
[97] J.-L., Colliot-Thélène and J.-J., Sansuc. La descente sur une variété rationnelle définie sur un corps de nombres. C. R. Acad. Sci. Paris Sér. A-B, 284(19):A1215–A1218, 1977.
[98] J.-L., Colliot-Thélène and J.-J., Sansuc. Variétés de première descente attachées aux variétés rationnelles. C. R. Acad. Sci. Paris Sér. A-B, 284(16):A967–A970, 1977.
[99] J.-L., Colliot-Thélène and J.-J., Sansuc. La descente sur les variétés rationnelles. In Journées de géometrie algébrique d'Angers, Juillet 1979/Algebraic geometry, Angers, 1979, pp. 223–237. Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.
[100] J.-L., Colliot-Thélène and J.-J., Sansuc. La descente sur les variétés rationnelles. II. Duke Math. J., 54(2):375–492, 1987.
[101] J.-L., Colliot-Thélène, J.-J., Sansuc, and P., Swinnerton-Dyer. Intersections of two quadrics and Châtelet surfaces. I. J. reine angew. Math., 373:37–107, 1987.
[102] J.-L., Colliot-Thélène, J.-J., Sansuc, and P., Swinnerton-Dyer. Intersections of two quadrics and Châtelet surfaces. II. J. reine angew. Math., 374:72–168, 1987.
[103] F. R., Cossec and I. V., Dolgachev. Enriques surfaces. I. Progress in Mathematics, Vol. 76. Birkhäuser Boston, Boston, 1989.
[104] D. A., Cox. The homogeneous coordinate ring of a toric variety. J. Algebraic Geom., 4(1):17–50, 1995.
[105] D. A., Cox, J. B., Little, and H. K., Schenck. Toric varieties. Graduate Studies in Mathematics, Vol. 124. American Mathematical Society, Providence, RI, 2011.
[106] C., De Concini and C., Procesi. Complete symmetric varieties. In Invariant theory (Montecatini, 1982). Lecture Notes in Mathematics, Vol. 996, pp. 1–44. Springer-Verlag, Berlin, 1983.
[107] J. A., De Loera, J., Rambau, and F., Santos. Triangulations. Algorithms and Computation in Mathematics, Vol. 25. Springer-Verlag, Berlin, 2010. Structures for algorithms and applications.
[108] M., Demazure. Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. (4), 3:507–588, 1970.
[109] U., Derenthal. Geometry of universal torsors. PhD thesis, Universität Göttingen, 2006.
[110] U., Derenthal. Universal torsors of del Pezzo surfaces and homogeneous spaces. Adv. Math., 213(2):849–864, 2007.
[111] U., Derenthal. Counting integral points on universal torsors. Int. Math. Res. Not. IMRN, (14):2648–2699, 2009.
[112] U., Derenthal. Manin's conjecture for a quintic del Pezzo surface with A2 singularity. arXiv:0710.1583, 2007.
[113] U., Derenthal. Singular Del Pezzo surfaces whose universal torsors are hypersur-faces. Proc. Lond. Math. Soc. (3), 108(3):638–681, 2014.
[114] U., Derenthal. Manin's conjecture for a certain singular cubic surface. arXiv:math.NT/0504016, 2005.
[115] U., Derenthal, A.-S., Elsenhans, and J., Jahnel. On the factor alpha in Peyre's constant. Math. Comp., 83(286):965–977, 2014.
[116] U., Derenthal and C., Frei. Counting imaginary quadratic points via universal torsors. Compositio Math., in press, arXiv:1302.6151, 2013.
[117] U., Derenthal and C., Frei. Counting imaginary quadratic points via universal torsors, II. Math. Proc. Cambridge Philos. Soc., 156(3):383–407, 2014.
[118] U., Derenthal and C., Frei. On Manin's conjecture for a certain singular cubic surface over imaginary quadratic fields. Int. Math. Res. Not. IMRN, in press, arXiv:1311.2809, 2013.
[119] U., Derenthal, M., Joyce, and Z., Teitler. The nef cone volume of generalized del Pezzo surfaces. Algebra Number Theory, 2(2):157–182, 2008.
[120] U., Derenthal and D., Loughran. Singular del Pezzo surfaces that are equivariant compactifications. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 377(Issledovaniya po Teorii Chisel. 10):26–43, 241, 2010.
[121] U., Derenthal and M., Pieropan. Cox rings over nonclosed fields, preprint, 2014.
[122] U., Derenthal, A., Smeets, and D., Wei. Universal torsors and values of quadratic polynomials represented by norms. Math. Ann., in press, arXiv:1202.3567, 2012.
[123] U., Derenthal and Yu., Tschinkel. Universal torsors over del Pezzo surfaces and rational points. In Equidistribution in number theory, an introduction. NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 237, pp. 169–196. Springer-Verlag, Dordrecht, 2007.
[124] K., Destagnol. La conjecture de Manin sur les surfaces de Châtelet, Master's thesis, Université Paris 7 Diderot, 2013.
[125] I. V., Dolgachev. Newton polyhedra and factorial rings. J. Pure Appl. Algebra, 18(3):253–258, 1980.
[126] I. V., Dolgachev. On automorphisms of Enriques surfaces. Invent. Math., 76(1):163–177, 1984.
[127] I. V., Dolgachev. Classical algebraic geometry. Cambridge University Press, Cambridge, 2012. A modern view.
[128] I. V., Dolgachev and Yi, Hu. Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math., (87):5–56, 1998.
[129] I. V., Dolgachev and De-Qi, Zhang. Coble rational surfaces. Amer. J. Math., 123(1):79–114, 2001.
[130] F., Donzelli. Algebraic density property of Danilov-Gizatullin surfaces. Math. Z., 272(3–4):1187–1194, 2012.
[131] J. J., Duistermaat. Discrete integrable systems. Springer Monographs in Mathematics. Springer-Verlag, New York, 2010. QRT maps and elliptic surfaces.
[132] D., Eisenbud. Commutative algebra. Graduate Texts in Mathematics, Vol. 150. Springer-Verlag, New York, 1995.
[133] D., Eisenbud and S., Popescu. The projective geometry of the Gale transform. J. Algebra, 230(1):127–173, 2000.
[134] E. J., Elizondo, K., Kurano, and K., Watanabe. The total coordinate ring of a normal projective variety. J. Algebra, 276(2):625–637, 2004.
[135] A.-S., Elsenhans and J., Jahnel. Moduli spaces and the inverse Galois problem for cubic surfaces. Trans. Amer. Math. Soc., in press, arXiv:1209.5591, 2012.
[136] G., Ewald. Polygons with hidden vertices. Beiträge Algebra Geom., 42(2):439–442, 2001.
[137] H., Flenner, S., Kaliman, and M., Zaidenberg. On the Danilov-Gizatullin isomorphism theorem. Enseign. Math. (2), 55(3-4):275–283, 2009.
[138] J., Franke, Yu. I., Manin, and Yu., Tschinkel. Rational points of bounded height on Fano varieties. Invent. Math., 95(2):421–435, 1989.
[139] C., Frei. Counting rational points over number fields on a singular cubic surface. Algebra Number Theory, 7(6):1451–1479, 2013.
[140] C., Frei and M., Pieropan. O-minimality on twisted universal torsors and Manin's conjecture over number fields. arXiv:1312.6603, 2013.
[141] W., Fulton. Introduction to toric varieties. Annals of Mathematics Studies, Vol. 131. Princeton University Press, Princeton, NJ, 1993.
[142] W., Fulton. Intersection theory. Ergebnisse der Mathematik und ihrer Grenzge-biete (3), Vol. 2, 2nd ed. Springer-Verlag, Berlin, 1998.
[143] W., Fulton and J., Harris. Representation theory. Graduate Texts in Mathematics, Vol. 129. Springer-Verlag, New York, 1991.
[144] G., Gagliardi. The Cox ring of a spherical embedding. J. Algebra, 397:548–569, 2014.
[145] S. A., Gaĭfullin. Affine toric SL(2)-embeddings. Mat. Sb., 199(3):3–24, 2008.
[146] C., Galindo and F., Monserrat. The total coordinate ring of a smooth projective surface. J. Algebra, 284(1):91–101, 2005.
[147] M. H., Gizatullin and V. I., Danilov. Automorphisms of affine surfaces. II. Izv. Akad. Nauk SSSR Ser. Mat., 41(1):54–103, 231, 1977.
[148] Y., Gongyo, S., Okawa, A., Sannai, and S., Takagi. Characterization of varieties of Fano type via singularities of Cox rings. J. Algebraic Geom., in press, arXiv:1201.1133, 2012.
[149] B., Green and T., Tao. Linear equations in primes. Ann. Math. (2), 171(3):1753–1850, 2010.
[150] F. D., Grosshans. Algebraic homogeneous spaces and invariant theory. Lecture Notes in Mathematics, Vol. 1673. Springer-Verlag, Berlin, 1997.
[151] A., Grothendieck. Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math., (8):222, 1961.
[152] N., Guay. Embeddings of symmetric varieties. Transform. Groups, 6(4):333–352, 2001.
[153] M., Hanselmann. Rational points on quartic hypersurfaces. PhD thesis, Ludwig-Maximilians-Universität München, 2012.
[154] D., Harari. Obstructions de Manin transcendantes. In Number theory (Paris, 1993–1994), London Mathematical Society Lecture Note Series, Vol. 235, pp. 75–87. Cambridge University Press, Cambridge, 1996.
[155] D., Harari. Groupes algébriques et points rationnels. Math. Ann., 322(4):811–826, 2002.
[156] D., Harari. Weak approximation on algebraic varieties. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progress in Mathematics, Vol. 226, pp. 43–60. Birkhäuser Boston, Boston, 2004.
[157] D., Harari and A. N., Skorobogatov. Descent theory for open varieties. In Torsors, etale homotopy and applications to rational points. London Mathematical Society Lecture Note Series, Vol. 405, pp. 250–279. Cambridge University Press, Cambridge, 2013.
[158] Y., Harpaz, A. N., Skorobogatov, and O., Wittenberg. The Hardy–Littlewood conjecture and rational points. Compositio Math., in press, arXiv:1304.3333, 2013.
[159] R., Hartshorne. Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156. Springer-Verlag, Berlin, 1970.
[160] R., Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, Vol. 52. Springer-Verlag, New York, 1977.
[161] H., Hasse. Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassen-körpertheorie im Kleinen. J. reine angew. Math., 162:145–154, 1930.
[162] B., Hassett and Yu., Tschinkel. Universal torsors and Cox rings. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002). Progress in Mathematics, Vol. 226, pp. 149–173. Birkhäuser Boston, Boston, 2004.
[163] J., Hausen. Equivariant embeddings into smooth toric varieties. Canad. J. Math., 54(3):554–570, 2002.
[164] J., Hausen. Geometric invariant theory based on Weil divisors. Compos. Math., 140(6):1518–1536, 2004.
[165] J., Hausen. Cox rings and combinatorics. II. Mosc. Math. J., 8(4):711–757, 847, 2008.
[166] J., Hausen and E., Herppich. Factorially graded rings of complexity one. In Torsors, etale homotopy and applications to rational points. London Mathematical Society Lecture Note Series, Vol. 405, pp. 414–428. Cambridge University Press, Cambridge, 2013.
[167] J., Hausen, E., Herppich, and H., Süß. Multigraded factorial rings and Fano varieties with torus action. Doc. Math., 16:71–109, 2011.
[168] J., Hausen, S., Keicher, and A., Laface. Computing Cox rings. arXiv:1305.4343, 2013.
[169] J., Hausen and H., Sufi. The Cox ring of an algebraic variety with torus action. Adv. Math., 225(2):977–1012, 2010.
[170] D. R., Heath-Brown. The density of rational points on Cayley's cubic surface. In Proceedings of the Session in Analytic Number Theory and Diophantine Equations. Bonner Mathematische Schriften, Vol. 360, pp. 33, Bonn, 2003. Universität Bonn.
[171] D. R., Heath-Brown. Zeros of pairs of quadratic forms. arXiv:1304.3894, 2013.
[172] D. R., Heath-Brown and A., Skorobogatov. Rational solutions of certain equations involving norms. Acta Math., 189(2):161–177, 2002.
[173] H., Hironaka. Triangulations of algebraic sets. In Algebraic geometry (Humboldt State Univ., Arcata, Calif., 1974). Proc. Sympos. Pure Math., Vol. 29, pp. 165–185. Amer. Math. Soc., Providence, RI, 1975.
[174] M., Hochster and J. L., Roberts. Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Adv. Math., 13:115–175, 1974.
[175] C., Hooley. On nonary cubic forms. J. reine angew. Math., 386:32–98, 1988.
[176] Y., Hu and S., Keel. Mori dream spaces and GIT. Michigan Math. J., 48:331–348, 2000.
[177] E., Huggenberger. Fano varieties with a torus action of complexity one. PhD thesis, Universiät Tübingen, 2013.
[178] J. E., Humphreys. Linear algebraic groups. Graduate Texts in Mathematics, Vol. 21. Springer-Verlag, New York, 1975.
[179] D., Hwang and J., Park. Redundant blow-ups and Cox rings of rational surfaces. arXiv:1303.2274, 2013.
[180] M.-N., Ishida. Graded factorial rings of dimension 3 of a restricted type. J. Math. Kyoto Univ., 17(3):441–456, 1977.
[181] S.-Y., Jow. A Lefschetz hyperplane theorem for Mori dream spaces. Math. Z., 268(1-2):197–209, 2011.
[182] T., Kajiwara. The functor of a toric variety with enough invariant effective Cartier divisors. Tohoku Math. J. (2), 50(1):139–157, 1998.
[183] Y., Kawamata and S., Okawa. Mori dream spaces of Calabi-Yau type and the log canonicity of the Cox rings. J. reine angew. Math., in press, arXiv:1202.2696, 2012.
[184] A. D., King. Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2), 45(180):515–530, 1994.
[185] S. L., Kleiman. Toward a numerical theory of ampleness. Ann. of Math. (2), 84:293–344, 1966.
[186] P., Kleinschmidt. A classification of toric varieties with few generators. Aequationes Math., 35(2-3):254–266, 1988.
[187] F., Knop. The Luna-Vust theory of spherical embeddings. In Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), pp. 225–249, Madras, 1991. Manoj Prakashan.
[188] F., Knop. Über Hilberts vierzehntes Problem für Varietäten mit Kompliziertheit eins. Math. Z., 213(1):33–36, 1993.
[189] F., Knop, H., Kraft, D., Luna, and T., Vust. Local properties of algebraic group actions. In Algebraische Transformationsgruppen und Invariantentheorie. DMV Seminar, Vol. 13, pp. 63–75. Birkhäuser, Basel, 1989.
[190] F., Knop, H., Kraft, and T., Vust. The Picard group of a G-variety. In Algebraische Transformationsgruppen und Invariantentheorie. DMV Seminar, Vol. 13, pp. 77–87. Birkhäuser, Basel, 1989.
[191] M., Koitabashi. Automorphism groups of generic rational surfaces. J. Algebra, 116(1):130–142, 1988.
[192] J., Kollár and S., Mori. Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, Vol. 134. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti.
[193] J., Kollár and E., Szabó. Fixed points of group actions and rational maps. arXiv:math/9905053, 1999.
[194] S., Kondō. Enriques surfaces with finite automorphism groups. Jpn. J. Math., 12(2):191–282, 1986.
[195] S., Kondō. Algebraic K3 surfaces with finite automorphism groups. Nagoya Math. J., 116:1–15, 1989.
[196] M., Koras and P., Russell. Linearization problems. In Algebraic group actions and quotients, pp. 91–107. Hindawi, Cairo, 2004.
[197] H., Kraft. Geometrische Methoden in der Invariantentheorie. Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984.
[198] A., Laface and D., Testa. Nef and semiample divisors on rational surfaces. In Torsors, étale homotopy and applications to rational points. London Mathematical Society Lecture Note Series, Vol. 405, pp. 429–446. Cambridge University Press, Cambridge, 2013.
[199] A., Laface and M., Velasco. Picard-graded Betti numbers and the defining ideals of Cox rings. J. Algebra, 322(2):353–372, 2009.
[200] A., Laface and M., Velasco. A survey on Cox rings. Geom. Dedicata, 139:269–287, 2009.
[201] K. F., Lai and K. M., Yeung. Rational points in flag varieties over function fields. J. Number Theory, 95(2):142–149, 2002.
[202] R., Lazarsfeld. Positivity in algebraic geometry. I. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 48. Springer-Verlag, Berlin, 2004.
[203] P., Le Boudec. Manin's conjecture for a cubic surface with 2A2 + A1 singularity type. Math. Proc. Cambridge Philos. Soc., 153(3):419–455, 2012.
[204] P., LeBoudec. Manin's conjecture for a quartic del Pezzo surface with A3 singularity and four lines. Monatsh. Math., 167(3-4):481–502, 2012.
[205] P., Le Boudec. Manin's conjecture for two quartic del Pezzo surfaces with 3A1 and A1 + A2 singularity types. Acta Arith., 151(2):109–163, 2012.
[206] P., Le Boudec. Affine congruences and rational points on a certain cubic surface. Algebra Number Theory, in press, arXiv:1207.2685, 2012.
[207] J., Lipman. Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math., (36):195–279, 1969.
[208] D., Loughran. Manin's conjecture for a singular quartic del Pezzo surface. J. Lond. Math. Soc. (2), 86(2):558–584, 2012.
[209] D., Loughran. Rational points of bounded height and the Weil restriction. Israel J. Math., in press, arXiv:1210.1792, 2012.
[210] D., Luna. Slices étales. In Sur les groupes algébriques, pages 81–105. Bull. Soc. Math. France, Paris, Mémoire 33. Soc. Math. France, Paris, 1973.
[211] D., Luna. Toute variéte magnifique est sphérique. Transform. Groups, 1(3):249–258, 1996.
[212] D., Luna and T., Vust. Plongements d'espaces homogènes. Comment. Math. Helv., 58(2):186–245, 1983.
[213] D., Maclagan and B., Sturmfels. Introduction to tropical geometry. In preparation.
[214] Yu. I., Manin. Cubic forms. North-Holland Mathematical Library, Vol. 4, 2nd ed. North-Holland Publishing Co., Amsterdam, 1986.
[215] K., Matsuki. Introduction to the Mori program. Universitext. Springer-Verlag, New York, 2002.
[216] H., Matsumura. Commutative ring theory. Cambridge Studies in Advanced Mathematics, Vol. 8, 2nd ed. Cambridge University Press, Cambridge, 1989.
[217] J., McKernan. Mori dream spaces. Jpn. J. Math., 5(1):127–151, 2010.
[218] E., Miller and B., Sturmfels. Combinatorial commutative algebra. Graduate Texts in Mathematics, Vol. 227. Springer-Verlag, New York, 2005.
[219] J. S., Milne. Étale cohomology. Princeton Mathematical Series, Vol. 33. Princeton University Press, Princeton, NJ, 1980.
[220] R., Miranda and U., Persson. On extremal rational elliptic surfaces. Math. Z., 193(4):537–558, 1986.
[221] S., Mori. Graded factorial domains. Jpn. J. Math., 3(2):223–238, 1977.
[222] D. R., Morrison. On K3 surfaces with large Picard number. Invent. Math., 75(1):105–121, 1984.
[223] S., Mukai. Geometric realization of T -shaped root systems and counterexamples to Hilbert's fourteenth problem. In Algebraic transformation groups and algebraic varieties. Encyclopaedia Mathematical Sciences, Vol. 132, pp. 123–129. Springer-Verlag, Berlin, 2004.
[224] D., Mumford. The red book of varieties and schemes. Lecture Notes in Mathematics, Vol. 1358, expanded edition. Springer-Verlag, Berlin, 1999.
[225] D., Mumford, J., Fogarty, and F., Kirwan. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34, 3rd ed. Springer-Verlag, Berlin, 1994.
[226] I. M., Musson. Differential operators on toric varieties. J. Pure Appl. Algebra, 95(3):303–315, 1994.
[227] M., Mustaţă. Vanishing theorems on toric varieties. Tohoku Math. J. (2), 54(3):451–470, 2002.
[228] M., Nagata. On the 14-th problem of Hilbert. Amer. J. Math., 81:766–772, 1959.
[229] N., Nakayama. Classification of log del Pezzo surfaces of index two. J. Math. Sci. Univ. Tokyo, 14(3):293–498, 2007.
[230] V. V., Nikulin. Quotient-groups of groups of automorphisms of hyperbolic forms of subgroups generated by 2-reflections. Dokl. Akad. Nauk SSSR, 248(6):1307–1309, 1979.
[231] V. V., Nikulin. K3 surfaces with a finite group of automorphisms and a Picard group of rank three. Trudy Mat. Inst. Steklov., 165:119–142, 1984. Algebraic geometry and its applications.
[232] V. V., Nikulin. A remark on algebraic surfaces with polyhedral Mori cone. Nagoya Math. J., 157:73–92, 2000.
[233] B., Nill. Complete toric varieties with reductive automorphism group. Math. Z., 252(4):767–786, 2006.
[234] T., Oda. Convex bodies and algebraic geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 15. Springer-Verlag, Berlin, 1988.
[235] T., Oda and H. S., Park. Linear Gale transforms and Gel'fand-Kapranov-Zelevinskij decompositions. Tohoku Math. J. (2), 43(3):375–399, 1991.
[236] A. L., Onishchik and È. B., Vinberg. Lie groups and algebraic groups. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990.
[237] P., Orlik and P., Wagreich. Isolated singularities of algebraic surfaces with C* action. Ann. of Math. (2), 93:205–228, 1971.
[238] P., Orlik and P., Wagreich. Algebraic surfaces with k*-action. Acta Math., 138(1–2):43–81, 1977.
[239] J. C., Ottem. On the Cox ring of P2 blown up in points on a line. Math. Scand., 109(1):22–30, 2011.
[240] J. C., Ottem. Cox rings of K3 surfaces with Picard number 2. J. Pure Appl. Algebra, 217(4):709–715, 2013.
[241] T. E., Panov. Toric Kempf–Ness sets. Proc. Steklov Inst. Math., 263:159–172, 2008.
[242] H. S., Park. The Chow rings and GKZ-decompositions for Q-factorial toric varieties. Tohoku Math. J. (2), 45(1):109–145, 1993.
[243] E., Peyre. Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J., 79(1):101–218, 1995.
[244] E., Peyre. Terme principal de la fonction zêta des hauteurs et torseurs universels. Astérisque, (251):259–298, 1998. Nombre et répartition de points de hauteur bornée (Paris, 1996).
[245] E., Peyre. Torseurs universels et méthode du cercle. In Rational points on algebraic varieties. Progress in Mathematics, Vol. 199, pp. 221–274. Birkhäuser, Basel, 2001.
[246] E., Peyre. Points de hauteur bornée et géométrie des variétés (d'après Y. Manin et al.). Astérisque, (282):Exp. No. 891, ix, 323–344, 2002. Séminaire Bourbaki, Vol. 2000/2001.
[247] E., Peyre. Points de hauteur bornée, topologie adélique et mesures de Tamagawa. J. Théor. Nombres Bordeaux, 15(1):319–349, 2003. Les XXIIèmes Journees Arithmetiques (Lille, 2001).
[248] E., Peyre. Counting points on varieties using universal torsors. In Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002). Progress in Mathematics, Vol. 226, pp. 61–81. Birkhäuser Boston, Boston, 2004.
[249] E., Peyre. Obstructions au principe de Hasse et à l'approximation faible. Astérisque, (299):Exp. No. 931, viii, 165–193, 2005. Séminaire Bourbaki. Vol. 2003/2004.
[250] E., Peyre. Points de hauteur bornée sur les variétés de drapeaux en caractéristique finie. Acta Arith., 152(2):185–216, 2012.
[251] I. I., Pjatecking-Šapiro and I. R., Šafarevič. Torelli's theorem for algebraic surfaces of type K3. Izv. Akad. Nauk SSSR Ser. Mat., 35:530–572, 1971.
[252] B., Poonen. Insufficiency of the Brauer-Manin obstruction applied to etale covers. Ann. of Math. (2), 171(3):2157–2169, 2010.
[253] B., Poonen. Rational points on varieties, http://www-math.mit.edu/~poonen/papers/Qpoints.pdf, 2008.
[254] O. N., Popov. The Cox ring of a Del Pezzo surface has rational singularities. arXiv:math.AG/0402154, 2004.
[255] V. L., Popov. Quasihomogeneous affine algebraic varieties of the group SL(2). Izv. Akad. Nauk SSSR Ser. Mat, 37:792–832, 1973.
[256] G. V., Ravindra and V., Srinivas. The Grothendieck-Lefschetz theorem for normal projective varieties. J. Algebraic Geom., 15(3):563–590, 2006.
[257] G. V., Ravindra and V., Srinivas. The Noether-Lefschetz theorem for the divisor class group. J. Algebra, 322(9):3373–3391, 2009.
[258] L., Renner. The cone of semi-simple monoids with the same factorial hull. arXiv:math.AG/0603222, 2006.
[259] A., Rittatore. Algebraic monoids and group embeddings. Transform. Groups, 3(4):375–396, 1998.
[260] A., Rittatore. Very flat reductive monoids. Publ. Mat. Urug., 9:93–121 (2002), 2001.
[261] P., Roberts. An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem. J. Algebra, 132(2):461–473, 1990.
[262] B., Saint-Donat. Projective models of K – 3 surfaces. Amer. J. Math., 96:602–639, 1974.
[263] P., Salberger. Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque, (251):91–258, 1998. Nombre et répartition de points de hauteur bornée (Paris, 1996).
[264] P., Samuel. Lectures on unique factorization domains. Notes by M. Pavman Murthy. Tata Institute of Fundamental Research Lectures on Mathematics, No. 30. Tata Institute of Fundamental Research, Bombay, 1964.
[265] S., Schanuel. Heights in number fields. Bull. Soc. Math. France, 107(4):433–449, 1979.
[266] G., Scheja and U., Storch. Zur Konstruktion faktorieller graduierter Integritätsbereiche. Arch. Math. (Basel), 42(1):45–52, 1984.
[267] V. V., Serganova. Torsors and representation theory of reductive groups. In Tor-sors, etale homotopy and applications to rational points. London Mathematical Society Lecture Note Series, Vol. 405, pp. 75–119. Cambridge University Press, Cambridge, 2013.
[268] V. V., Serganova and A. N., Skorobogatov. Del Pezzo surfaces and representation theory. Algebra Number Theory, 1(4):393–419, 2007.
[269] V. V., Serganova and A. N., Skorobogatov. On the equations for universal torsors over del Pezzo surfaces. J. Inst. Math. Jussieu, 9(1):203–223, 2010.
[270] V. V., Serganova and A. N., Skorobogatov. Adjoint representation of E8 and del Pezzo surfaces of degree 1. Ann. Inst. Fourier (Grenoble), 61(6):2337–2360 (2012), 2011.
[271] J.-P., Serre. Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble, 6:1–42, 1955–1956.
[272] J.-P., Serre. A course in arithmetic. Springer-Verlag, New York, 1973.
[273] J.-P., Serre. Local fields. Graduate Texts in Mathematics, Vol. 67. Springer-Verlag, New York, 1979.
[274] J.-P., Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, English edition, 2002.
[275] V. V., Shokurov. A nonvanishing theorem. Izv. Akad. Nauk SSSR Ser. Mat., 49(3):635–651, 1985.
[276] C. M., Skinner. Forms over number fields and weak approximation. Compositio Math., 106(1):11–29, 1997.
[277] A. N., Skorobogatov. On a theorem of Enriques-Swinnerton-Dyer. Ann. Fac. Sci. Toulouse Math. (6), 2(3):429–440, 1993.
[278] A. N., Skorobogatov. Beyond the Manin obstruction. Invent. Math., 135(2):399–424, 1999.
[279] A. N., Skorobogatov. Torsors and rational points. Cambridge Tracts in Mathematics, Vol. 144. Cambridge University Press, Cambridge, 2001.
[280] T. A., Springer. Linear algebraic groups, 2nd ed. Modern Birkhäuser Classics. Birkhäuser Boston, Boston, 2009.
[281] M., Stillman, D., Testa, and M., Velasco. Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces. J. Algebra, 316(2):777–801, 2007.
[282] B., Sturmfels. Gröbner bases and convex polytopes. University Lecture Series, Vol. 8. American Mathematical Society, Providence, RI, 1996.
[283] B., Sturmfels and M., Velasco. Blow-ups of & Popf;n – 3 at n points and spinor varieties. J. Commut. Algebra, 2(2):223–244, 2010.
[284] B., Sturmfels and Z., Xu. Sagbi bases of Cox-Nagata rings. J. Eur. Math. Soc. (JEMS), 12(2):429–459, 2010.
[285] H., Sumihiro. Equivariant completion. J. Math. Kyoto Univ., 14:1–28, 1974.
[286] H., Süß. Canonical divisors on T-varieties. arXiv:0811.0626, 2008.
[287] M., Swarbrick Jones. A Note On a Theorem of Heath-Brown and Skorobogatov. Q. J. Math. 64(4):1239–1251, 2013.
[288] J., Święcicka. Quotients of toric varieties by actions of subtori. Colloq. Math., 82(1):105–116, 1999.
[289] J., Święcicka. A combinatorial construction of sets with good quotients by an action of a reductive group. Colloq. Math., 87(1):85–102, 2001.
[290] P., Swinnerton-Dyer. Two special cubic surfaces. Mathematika, 9:54–56, 1962.
[291] P., Swinnerton-Dyer. The solubility of diagonal cubic surfaces. Ann. Sci. École Norm. Sup. (4), 34(6):891–912, 2001.
[292] D., Testa, A., Várilly-Alvarado, and M., Velasco. Cox rings of degree one del Pezzo surfaces. Algebra Number Theory, 3(7):729–761, 2009.
[293] D., Testa, A., Várilly-Alvarado, and M., Velasco. Big rational surfaces. Math. Ann., 351(1):95–107, 2011.
[294] J., Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129(4):1087–1104, 2007.
[295] M., Thaddeus. Geometric invariant theory and flips. J. Amer. Math. Soc., 9(3):691–723, 1996.
[296] D. A., Timashev. Homogeneous spaces and equivariant embeddings. Invariant Theory and Algebraic Transformation Groups, 8. Encyclopaedia of Mathematical Sciences, Vol. 138. Springer-Verlag, Heidelberg, 2011.
[297] B., Totaro. The cone conjecture for Calabi-Yau pairs in dimension 2. Duke Math. J., 154(2):241–263, 2010.
[298] É. B., Vinberg. Complexity of actions of reductive groups. Funktsional. Anal. i Prilozhen., 20(1):1–13, 96, 1986.
[299] É. B., Vinberg. On reductive algebraic semigroups. In Lie groups and Lie algebras: E. B. Dynkin's Seminar. American Mathematical Society Translations Series 2, Vol. 169, pp. 145–182. American Mathematical Society, Providence, RI, 1995.
[300] É. B., Vinberg. Classification of 2-reflective hyperbolic lattices of rank 4. Tr. Mosk. Mat. Obs., 68:44–76, 2007.
[301] É. B., Vinberg and V. L., Popov. Invariant theory. In Algebraic Geometry, IV. Encyclopedia of Mathematical Sciences, Vol. 55, pp. 123–278. Springer-Verlag, Berlin, 1994.
[302] V. E., Voskresenski. Algebraicheskie tory. Izdat. “Nauka,” Moscow, 1977.
[303] V. E., Voskresenski. Algebraic groups and their birational invariants. Translations of Mathematical Monographs, Vol. 179. American Mathematical Society, Providence, RI, 1998.
[304] W. C., Waterhouse. Introduction to affine group schemes. Graduate Texts in Mathematics, Vol. 66. Springer-Verlag, New York, 1979.
[305] R., Wazir. Arithmetic on elliptic threefolds. Compos. Math., 140(3):567–580, 2004.
[306] D., Wei. On the equation NK/k(Ξ) = P(t). Proc. London Math. Soc. (3), in press, arXiv:1202.4115, 2012.
[307] O., Wittenberg. Intersections de deux quadriques et pinceaux de courbes de genre 1/Intersections of two quadrics and pencils of curves of genus 1. Lecture Notes in Mathematics, Vol. 1901. Springer-Verlag, Berlin, 2007.
[308] J., Wlodarczyk. Embeddings in toric varieties and prevarieties. J. Algebraic Geom., 2(4):705–726, 1993.
[309] J., Wlodarczyk. Maximal quasiprojective subsets and the Kleiman-Chevalley quasiprojectivity criterion. J. Math. Sci. Univ. Tokyo, 6(1):41–47, 1999.
[310] D.-Q., Zhang. Quotients of K3 surfaces modulo involutions. Jpn. J. Math., 24(2):335–366, 1998.
[311] V. S., Zhgun. On embeddings of universal torsors over del Pezzo surfaces into cones over flag varieties. Izv. Ross. Akad. Nauk Ser. Mat., 74(5):3–44, 2010.

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