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ON TORUS ACTIONS OF HIGHER COMPLEXITY

Part of: Algebraic groups Special varieties Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  31 October 2019

JÜRGEN HAUSEN ,
CHRISTOFF HISCHE  and
MILENA WROBEL
JÜRGEN HAUSEN
Affiliation:
Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany; juergen.hausen@uni-tuebingen.de, hische@math.uni-tuebingen.de
CHRISTOFF HISCHE
Affiliation:
Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany; juergen.hausen@uni-tuebingen.de, hische@math.uni-tuebingen.de
MILENA WROBEL
Affiliation:
Institut für Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany; milena.wrobel@uni-oldenburg.de

Abstract

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We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of ‘general arrangement varieties’ and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.

MSC classification

Primary: 14L30: Group actions on varieties or schemes (quotients) 14M25: Toric varieties, Newton polyhedra 14J45: Fano varieties
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e38
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Altmann, K. and Hausen, J., ‘Polyhedral divisors and algebraic torus actions’, Math. Ann. 334(3) (2006), 557–607.Google Scholar
Altmann, K., Hausen, J. and Süss, H., ‘Gluing affine torus actions via divisorial fans’, Transform. Groups 13(2) (2008), 215–242.Google Scholar
Altmann, K. and Hein, G., ‘A fansy divisor on M 0, n ’, J. Pure Appl. Algebra 212(4) (2008), 840–850.Google Scholar
Arzhantsev, I., ‘On the factoriality of Cox rings’, Mat. Zametki 85(5) (2009), 643–651.Google Scholar
Arzhantsev, I., Derenthal, U., Hausen, J. and Laface, A., Cox Rings, Cambridge Studies in Advanced Mathematics 144 (Cambridge University Press, Cambridge, 2015).Google Scholar
Audin, M., The Topology of Torus Actions on Symplectic Manifolds, Progress in Mathematics, 93 (Birkhäuser, Basel, 1991), Translated from the French by the author.Google Scholar
Bäker, H., Hausen, J. and Keicher, S., ‘On Chow quotients of torus actions’, Michigan Math. J. 64(3) (2015), 451–473.Google Scholar
Batyrev, V. V., ‘Toric Fano threefolds’, Izv. Akad. Nauk SSSR Ser. Mat. 45(4) (1981), 704–717. 927.Google Scholar
Batyrev, V. V., ‘On the classification of toric Fano 4-folds’, J. Math. Sci. (New York) 94(1) (1999), 1021–1050.Google Scholar
Batyrev, V. V., ‘Toric degenerations of Fano varieties and constructing mirror manifolds’, inThe Fano Conference (Univ. Torino, Turin, 2004), 109–122.Google Scholar
Bechtold, B., ‘Factorially graded rings and Cox rings’, J. Algebra 369 (2012), 351–359.Google Scholar
Bechtold, B., ‘Valuative and geometric characterizations of Cox sheaves’, J. Commut. Algebra 10(1) (2018), 1–43.Google Scholar
Bechtold, B., Hausen, J., Huggenberger, E. and Nicolussi, M., ‘On terminal Fano 3-folds with 2-torus action’, Int. Math. Res. Not. IMRN 5 (2016), 1563–1602.Google Scholar
Berchtold, F. and Hausen, J., ‘Homogeneous coordinates for algebraic varieties’, J. Algebra 266(2) (2003), 636–670.Google Scholar
Berchtold, F. and Hausen, J., ‘Cox rings and combinatorics’, Trans. Amer. Math. Soc. 359(3) (2007), 1205–1252.Google Scholar
Birkar, C., ‘Singularities of linear systems and boundedness of Fano varieties’, Preprint, 2016, arXiv:1609.05543.Google Scholar
Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., ‘Existence of minimal models for varieties of log general type’, J. Amer. Math. Soc. 23(2) (2010), 405–468.Google Scholar
Borelli, M., ‘Divisorial varieties’, Pacific J. Math. 13 (1963), 375–388.Google Scholar
Bourqui, D., ‘La conjecture de Manin géométrique pour une famille de quadriques intrinsèques’, Manuscripta Math. 135(1–2) (2011), 1–41.Google Scholar
Casagrande, C., ‘On the birational geometry of Fano 4-folds’, Math. Ann. 355(2) (2013), 585–628.Google Scholar
Castravet, A.-M. and Tevelev, J., ‘ M 0, n is not a Mori dream space’, Duke Math. J. 164(8) (2015), 1641–1667.Google Scholar
Chow, W.-L., ‘On the geometry of algebraic homogeneous spaces’, Ann. of Math. (2) 50 (1949), 32–67.Google Scholar
Colliot-Thélène, J.-L. and Sansuc, J.-J., ‘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(18) (1976), A1113–A1116. Aii.Google Scholar
Cox, D. A., ‘The homogeneous coordinate ring of a toric variety’, J. Algebraic Geom. 4(1) (1995), 17–50.Google Scholar
Cox, D. A., Little, J. B. and Schenck, H. K., Toric Varieties, Graduate Studies in Mathematics, 124 (American Mathematical Society, Providence, RI, 2011).Google Scholar
Craw, A. and Maclagan, D., ‘Fiber fans and toric quotients’, Discrete Comput. Geom. 37(2) (2007), 251–266.Google Scholar
Danilov, V. I., ‘The geometry of toric varieties’, Uspekhi Mat. Nauk 33(2(200)) (1978), 85–134. 247.Google Scholar
Demazure, M., ‘Sous-groupes algébriques de rang maximum du groupe de Cremona’, Ann. Sci. Éc. Norm. Supér. (4) 3 (1970), 507–588.Google Scholar
Elizondo, E. J., Kurano, K. and Watanabe, K.-I., ‘The total coordinate ring of a normal projective variety’, J. Algebra 276(2) (2004), 625–637.Google Scholar
Fahrner, A. and Hausen, J., ‘On intrinsic quadrics, Canad. J. Math. to appear’, Preprint, 2017, arXiv:1712.09822, doi:10.4153/CJM-2018-037-5.Google Scholar
Fahrner, A., Hausen, J. and Nicolussi, M., ‘Smooth projective varieties with a torus action of complexity 1 and Picard number 2’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18(2) (2018), 611–651.Google Scholar
Fieseler, K.-H. and Kaup, L., ‘Fixed points, exceptional orbits, and homology of affine C ∗ -surfaces’, Compos. Math. 78(1) (1991), 79–115.Google Scholar
Fieseler, K.-H. and Kaup, L., ‘On the geometry of affine algebraic C ∗ -surfaces’, inProblems in the Theory of Surfaces and their Classification (Cortona, 1988), Sympos. Math., XXXII (Academic Press, London, 1991), 111–140.Google Scholar
Flenner, H. and Zaidenberg, M., ‘Normal affine surfaces with ℂ∗ -actions’, Osaka J. Math. 40(4) (2003), 981–1009.Google Scholar
Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies 131 (Princeton University Press, Princeton, NJ, 1993), The William H. Roever Lectures in Geometry.Google Scholar
González, J. L. and Karu, K., ‘Some non-finitely generated Cox rings’, Compos. Math. 152(5) (2016), 984–996.Google Scholar
Hassett, B. and Tschinkel, Y., ‘Universal torsors and Cox rings’, inArithmetic of Higher-dimensional Algebraic Varieties (Palo Alto, CA, 2002), Progress in Mathematics, 226 (Birkhäuser Boston, Boston, MA, 2004), 149–173.Google Scholar
Hausen, J., ‘Equivariant embeddings into smooth toric varieties’, Canad. J. Math. 54(3) (2002), 554–570.Google Scholar
Hausen, J., ‘Producing good quotients by embedding into toric varieties’, inGeometry of Toric Varieties, Sémin. Congr., 6 (Soc. Math. France, Paris, 2002), 193–212.Google Scholar
Hausen, J., ‘Cox rings and combinatorics. II’, Mosc. Math. J. 8(4) (2008), 711–757. 847.Google Scholar
Hausen, J. and Herppich, E., ‘Factorially graded rings of complexity one’, inTorsors, étale Homotopy and Applications to Rational Points, London Mathematical Society Lecture Note Series, 405 (Cambridge University Press, Cambridge, 2013), 414–428.Google Scholar
Hausen, J., Herppich, E. and Süss, Hendrik, ‘Multigraded factorial rings and Fano varieties with torus action’, Doc. Math. 16 (2011), 71–109.Google Scholar
Hausen, J. and Keicher, S., ‘A software package for Mori dream spaces’, LMS J. Comput. Math. 18(1) (2015), 647–659.Google Scholar
Hausen, J., Keicher, S. and Laface, A., ‘Computing Cox rings’, Math. Comp. 85(297) (2016), 467–502.Google Scholar
Hausen, J., Keicher, S. and Laface, A., ‘On blowing up the weighted projective plane’, Math. Z. 290(3–4) (2018), 1339–1358.Google Scholar
Hausen, J. and Süß, H., ‘The Cox ring of an algebraic variety with torus action’, Adv. Math. 225(2) (2010), 977–1012.Google Scholar
Hausen, J. and Wrobel, M., ‘Non-complete rational T-varieties of complexity one’, Math. Nachr. 290(5-6) (2017), 815–826.Google Scholar
Hu, Y. and Keel, S., ‘Mori dream spaces and GIT’, Michigan Math. J. 48(1) (2000), 331–348. Dedicated to William Fulton on the occasion of his 60th birthday.Google Scholar
Iskovskih, V. A., ‘Fano threefolds. I’, Izv. Akad. Nauk SSSR Ser. Mat. 41(3) (1977), 516–562. 717.Google Scholar
Iskovskih, V. A., ‘Fano threefolds. II’, Izv. Akad. Nauk SSSR Ser. Mat. 42(3) (1978), 506–549.Google Scholar
Kapranov, M. M., ‘Chow quotients of Grassmannians. I’, inI. M. Gel’fand Seminar, Adv. Soviet Math., 16 (American Mathematical Society, Providence, RI, 1993), 29–110.Google Scholar
Kapranov, M. M., Sturmfels, B. and Zelevinsky, A. V., ‘Quotients of toric varieties’, Math. Ann. 290(4) (1991), 643–655.Google Scholar
Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal Embeddings. I, Lecture Notes in Mathematics, 339 (Springer, Berlin–New York, 1973).Google Scholar
Kreuzer, M. and Nill, B., ‘Classification of toric Fano 5-folds’, Adv. Geom. 9(1) (2009), 85–97.Google Scholar
Luna, D., ‘Slices étales’, inSur les groupes algébriques, Mémoire, 33 (Bulletin de la Société Mathématique de France, Paris, 1973), 81–105.Google Scholar
Luna, D. and Vust, T., ‘Plongements d’espaces homogènes’, Comment. Math. Helv. 58(2) (1983), 186–245.Google Scholar
Milne, J. S., Étale Cohomology, Princeton Mathematical Series, 33 (Princeton University Press, Princeton, NJ, 1980).Google Scholar
Mori, S., ‘Graded factorial domains’, Japan. J. Math. (N.S.) 3(2) (1977), 223–238.Google Scholar
Mori, S. and Mukai, S., ‘Classification of Fano 3-folds with B 2⩾2’, Manuscripta Math. 36(2) (1981/82), 147–162.Google Scholar
Oda, T., Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15 (Springer, Berlin, 1988), An introduction to the theory of toric varieties, Translated from the Japanese.Google Scholar
Orlik, P. and Wagreich, P., ‘Isolated singularities of algebraic surfaces with C ∗ action’, Ann. of Math. (2) 93 (1971), 205–228.Google Scholar
Orlik, P. and Wagreich, P., ‘Singularities of algebraic surfaces with C ∗ action’, Math. Ann. 193 (1971), 121–135.Google Scholar
Orlik, P. and Wagreich, P., ‘Algebraic surfaces with k ∗ -action’, Acta Math. 138(1–2) (1977), 43–81.Google Scholar
Pinkham, H. C., ‘Normal surface singularities with C ∗ action’, Math. Ann. 227(2) (1977), 183–193.Google Scholar
Rosenlicht, M., ‘A remark on quotient spaces’, An. Acad. Brasil. Ci. 35 (1963), 487–489.Google Scholar
Samuel, P., 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).Google Scholar
Timashev, D. A., ‘Torus actions of complexity one’, inToric Topology, Contemporary Mathematics, 460 (American Mathematical Society, Providence, RI, 2008), 349–364.Google Scholar
Timashëv, D. A., ‘ G-manifolds of complexity 1’, Uspekhi Mat. Nauk 51(3(309)) (1996), 213–214.Google Scholar
Włodarczyk, J., ‘Embeddings in toric varieties and prevarieties’, J. Algebraic Geom. 2(4) (1993), 705–726.Google Scholar
Wrobel, M., ‘Structural properties of Cox rings of $T$ -varieties’, Doctoral Dissertation, Universität Tübingen, 2018. https://publikationen.uni-tuebingen.de.Google Scholar