Let G be one of the groups SLn(ℂ), Sp2n (ℂ), SOm(ℂ), Om(ℂ), or G 2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙN is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.
@article{bwmeta1.element.doi-10_2478_s11533-011-0097-9,
author = {Fedor Bogomolov and Christian B\"ohning and Hans-Christian Graf von Bothmer},
title = {Linear bounds for levels of stable rationality},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {466-520},
zbl = {1247.14010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0097-9}
}
Fedor Bogomolov; Christian Böhning; Hans-Christian Graf von Bothmer. Linear bounds for levels of stable rationality. Open Mathematics, Tome 10 (2012) pp. 466-520. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0097-9/
[1] Antonyan L.V., Classification of four-vectors of an eight-dimensional space, Trudy Sem. Vektor. Tenzor. Anal., 1981, 20, 144–161 (in Russian) | Zbl 0467.15018
[2] Beauville A., Determinantal hypersurfaces, Michigan Math. J., 2000, 48, 39–64 http://dx.doi.org/10.1307/mmj/1030132707 | Zbl 1076.14534
[3] Bogomolov F.A., Stable rationality of quotient varieties by simply connected groups, Mat. Sb. (N.S.), 1986, 130(1), 3–17 (in Russian) | Zbl 0615.14031
[4] Bogomolov F.A., Rationality of the moduli of hyperelliptic curves of arbitrary genus, In: Conference in Algebraic Geometry, Vancouver, 1984, CMS Conf. Proc., 6, American Mathematical Society, Providence, 1986, 17–37
[5] Bogomolov F., Böhning Chr., Graf von Bothmer H.-Chr., Rationality of quotients by linear actions of affine groups, Sci. China Math. (in press), DOI: 10.1007/s11425-010-4127-z | Zbl 1237.14026
[6] Bogomolov F.A., Katsylo P.I., Rationality of some quotient varieties, Mat. Sb. (N.S.), 1985, 126(4), 584–589 (in Russian) | Zbl 0591.14040
[7] Dolgachev I.V., Rationality of fields of invariants, In: Algebraic Geometry, Brunswick, 1985, Proc. Sympos. Pure Math., 46(2), American Mathematical Society, Providence, 1987, 3–16
[8] Fulton W., Harris J., Representation Theory, Grad. Texts in Math., 129, Springer, New York, 1991 http://dx.doi.org/10.1007/978-1-4612-0979-9
[9] Gille P., Szamuely T., Central Simple Algebras and Galois Cohomology, Cambridge Stud. Adv. Math., 101, Cambridge University Press, Cambridge, 2006 http://dx.doi.org/10.1017/CBO9780511607219 | Zbl 1137.12001
[10] Goodman R., Wallach N.R., Symmetry, Representations, and Invariants, Grad. Texts in Math., 255, Springer, Dordrecht, 2009 http://dx.doi.org/10.1007/978-0-387-79852-3
[11] Hoffmann N., Moduli stacks of vector bundles on curves and the King-Schofield rationality proof, In: Cohomological and Geometric Approaches to Rationality Problems, Progr. Math., 282, Birkhäuser, Boston, 2010, 133–148 http://dx.doi.org/10.1007/978-0-8176-4934-0_5 | Zbl 1203.14038
[12] Katanova A.A., Explicit form of certain multivector invariants, In: Lie Groups, their Discrete Subgroups, and Invariant Theory, Adv. Soviet Math., 8, American Mathematical Society, Providence, 1992, 87–93 | Zbl 0748.15028
[13] Katsylo P.I., Rationality of orbit spaces of irreducible representations of the group SL2, Izv. Akad. Nauk SSSR Ser. Mat., 1983, 47(1), 26–36 (in Russian)
[14] Katsylo P.I., Rationality of the moduli spaces of hyperelliptic curves, Izv. Akad. Nauk SSSR Ser. Mat., 1984, 48(4), 705–710 (in Russian)
[15] King A., Schofield A., Rationality of moduli of vector bundles on curves, Indag. Math. (N.S.), 1999, 10(4), 519–535 http://dx.doi.org/10.1016/S0019-3577(00)87905-7 | Zbl 1043.14502
[16] Miyata T., Invariants of certain groups. I, Nagoya Math. J., 1971, 41, 69–73 | Zbl 0211.06801
[17] Perroud M., On the irreducible representations of the Lie algebra chain G 2 ⊃ A 2, J. Mathematical Phys., 1976, 17(11), 1998–2006 http://dx.doi.org/10.1063/1.522839 | Zbl 0346.17007
[18] Popov A.M., Irreducible semisimple linear Lie groups with finite stationary subgroups of general position, Funct. Anal. Appl., 1978, 12(2), 154–155 http://dx.doi.org/10.1007/BF01076269 | Zbl 0404.22018
[19] Popov V.L., Classification of spinors in dimension fourteen, Trans. Moscow Math. Soc., 1980, 1, 181–232 | Zbl 0443.20038
[20] Popov V.L., Vinberg E.B., Invariant Theory, In: Algebraic Geometry IV, Encyclopaedia Math. Sci., 55, Springer, Berlin, 1994
[21] Saltman D., Lectures on Division Algebras, CBMS Regional Conf. Ser. in Math., 94, American Mathematical Society, Providence, 1999 | Zbl 0934.16013
[22] Schwarz G.W., Brion M., Théorie des Invariants et Géométrie des Variétés Quotients, Travaux en Cours, 61, Hermann Éditeurs des Sciences et des Arts, Paris, 2000
[23] Springer T.A., Veldkamp F.D., Octonions, Jordan Algebras and Exceptional Groups, Springer Monogr. Math., Springer, Berlin, 2000 | Zbl 1087.17001
[24] Van den Bergh M., The center of the generic division algebra, J. Algebra, 1989, 127(1), 106–126 http://dx.doi.org/10.1016/0021-8693(89)90277-9
[25] Vinberg E.B., The Weyl group of a graded Lie algebra, Math. USSR-Izv., 1976, 10(3), 463–495 http://dx.doi.org/10.1070/IM1976v010n03ABEH001711 | Zbl 0371.20041
[26] Wallach N., Yacobi O., A multiplicity formula for tensor products of SL2 modules and an explicit Sp2n to Sp2n−2×Sp2 branching formula, In: Symmetry in Mathematics and Physics, Contemp. Math., 490, American Mathematical Society, Providence, 2009, 151–155 http://dx.doi.org/10.1090/conm/490/09592