Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T16:02:39.622Z Has data issue: false hasContentIssue false
  • English
  • Français

Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties

Published online by Cambridge University Press:  20 November 2018

Zinovy Reichstein  and
Boris Youssin
Zinovy Reichstein
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331-4506, U.S.A. email: [email protected]
Boris Youssin
Affiliation:
Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety ${{X}^{'}}$ with the following property: the stabilizer of every point of ${{X}^{'}}$ is isomorphic to a semidirect product $U\rtimes A$ of a unipotent group $U$ and a diagonalizable group $A$.

As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

Keywords

14L3014E1514E0512E0520G10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 5 , 01 October 2000 , pp. 1018 - 1056
Copyright
Copyright © Canadian Mathematical Society 2000

References

References

[A] Adams, J. F., 2-tori in E 8. Math. Ann. 287(1987), 29–39.Google Scholar
[BM1] Bierstone, E. and Milman, P. D., A simple constructive proof of canonical resolution of singularities. Effective methods in algebraic geometry, Progress in Math. 94, Birkhäuser, Boston, 1991.Google Scholar
[BM2] Bierstone, E. and Milman, P. D., Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. (2) 128(1997), 207–302.Google Scholar
[Bo] Borel, A., Sous groupes commutatifs et torsion des groupes de Lie compacts connexes. TôhokuMath. J. (2) 13(1961), 216–240.Google Scholar
[BS] Borel, A. and Serre, J.-P., Sur certains sous groupes des groupes de Lie compacts. Comment. Math. Helv. 27(1953), 128–139.Google Scholar
[BR1] Buhler, J. and Reichstein, Z., On the essential dimension of a finite group. Compositio Math. 106(1997), 159–179.Google Scholar
[BR2] Buhler, J. and Reichstein, Z., On Tschirnhaus transformations. In: Number Theory, Proceedings of a conference held at Penn. State University (eds. S. Ahlgren, G. Andrews and K. Ono), Kluwer Acad. Publishers, 127–142. Preprint available at http://ucs.orst.edu/~reichstz/pub.html.Google Scholar
[CS] Cohen, A. M. and Seitz, G. M., The r-rank of the groups of exceptional Lie type. Proceedings of the Konink. Nederl. Akad. Wetensc. Ser. A (3) 90(1997), 251–259.Google Scholar
[Ga] Garibaldi, R. S., Structurable algebras and groups of type E6 and E7. Preprint.Google Scholar
[Gri] Griess, R. L., Jr. Elementary abelian p-subgroups of algebraic groups. Geom. Dedicata 39(1991), 253– 305.Google Scholar
[Gro] Grothendieck, A., La torsion homologique et les sections rationnelles. Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, 2nd année, IHP, 1958.Google Scholar
[EGA I] Grothendieck, A., Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4(1960).Google Scholar
[Ha] Hartshorne, R., Algebraic geometry. Springer, 1977.Google Scholar
[Hi] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II. Ann. of Math. 79(1964), 109–326.Google Scholar
[Hu] Humphreys, J. E.. Linear Algebraic Groups. Springer-Verlag, 1975.Google Scholar
[J1] Jacobson, N., The Theory of Rings. Math. Surveys Amer. Math. Soc., Providence, Rhode Island, 1943.Google Scholar
[J2] Jacobson, N., Structure and Representations of Jordan Algebras. Amer. Math. Soc., Providence, Rhode Island, 1968.Google Scholar
[KMRT] Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P., The Book of Involutions. Amer. Math. Soc. Colloquium Publications 44, 1998.Google Scholar
[Ko] Kordonsky, V. E., On essential dimension and Serre's Conjecture II for exceptional groups. In Russian, preprint.Google Scholar
[L] Lang, S., Algebra. Addison-Wesley, 1965.Google Scholar
[OV] Onishchik, A. L. and Vinberg, E. B., Lie Groups and Algebraic Groups. Springer-Verlag, 1990.Google Scholar
[P] Parusiński, A., Lipschitz properties of semianalytic sets. Ann. Inst. Fourier (Grenoble) 38(1988), 189–213.Google Scholar
[Po] Popov, V. L., Sections in Invariant Theory. Proceedings of the Sophus Lie Memorial Conference, Scandinavian Univ. Press, 1994, 315–362.Google Scholar
[PV] Popov, V. L. and Vinberg, E. B., Invariant Theory. in Encyclopaedia of Math. Sciences 55, Algebraic Geometry IV, (eds. A. N. Parshin and I. R. Shafarevich), Springer-Verlag, 1994.Google Scholar
[Re1] Reichstein, Z., On a theorem of Hermite and Joubert. Canad. J. Math. (1) 51(1999), 69–95.Google Scholar
[Re2] Reichstein, Z., On the notion of essential dimension for algebraic groups. Transformation Groups (3) 5(2000), to appear. Preprint available at http://ucs.orst.edu/~reichstz/pub.html.Google Scholar
[RY1] Reichstein, Z. and B. Youssin, Splitting fields of G-varieties. Pacific J. Math, to appear. Preprint available at http://ucs.orst.edu/~reichstz/pub.html.Google Scholar
[RY2] Reichstein, Z. and B. Youssin, Parusi´nski's lemma via algebraic geometry. Electronic Research Announcements of the Amer. Math. Soc. 5(1999), 136–145, http://www.ams.org/era/home-1999.html.Google Scholar
[RY3] Reichstein, Z. and B. Youssin, Equivariant resolution of points of indeterminacy. Preprint available at http://ucs.orst.edu/~reichstz/pub.html.Google Scholar
[Ro1] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math. 78(1956), 401–443.Google Scholar
[Ro2] Rosenlicht, M., A remark on quotient spaces. An. Acad. Brasil. Ciênc. 35(1963), 487–489.Google Scholar
[Rost1] Rost, M., Notes on 16-dimensional trace forms. Preprint, November 1998, http://www.physik.uni-regensburg.de/rom03516.Google Scholar
[Rost2] Rost, M., On Galois cohomology of Spin(14). Preprint, March 1999, http://www.physik.uni-regensburg.de/rom03516.Google Scholar
[RST] Rost, M., Serre, J.-P. and Tignol, J.-P., The trace form of a central simple algebra of degree four. In preparation. [Row] L. H. Rowen, Polynomial Identities in Ring Theory. Academic Press, 1980.Google Scholar
[Se1] Serre, J.-P., Espaces fibrés algébriques. Exposé 1, Séminaire C. Chevalley, Anneaux de Chow et applications, 2nd année, IHP, 1958.Google Scholar
[Se2] Serre, J.-P., Cohomologie galoisienne: progrès et problèmes. In: Séminaire Bourbaki, Volume 1993/94, Expos és 775–789, Astérisque 227(1995), 229–257.Google Scholar
[Se3] Serre, J.-P., Galois Cohomology, Springer, 1997.Google Scholar
[Se4] Serre, J.-P., letter from October 1, 1998.Google Scholar
[Se5] Serre, J.-P., letter from November 16, 1998.Google Scholar
[St] Steinberg, R., Generators, relations, and coverings of algebraic groups, II. J. Algebra 71(1981), 527–543.Google Scholar
[Su] Sumihiro, H., Equivariant completion. J. Math. KyotoUniv. (1) 14(1974), 1–28.Google Scholar
[V1] Villamayor, O. E. U.,Constructiveness of Hironaka's resolution. Ann. Sci. École. Norm. Sup. (4) 22(1989), 1–32.Google Scholar
[V2] Villamayor, O. E. U., Patching local uniformizations. Ann. Sci. École. Norm. Sup. (4) 25(1992), 629–677.Google Scholar
[Wo] Wood, J. A., Spinor groups and algebraic coding theory. J. Combin. Theory Ser. A 51(1989), 277–313. Google Scholar

References

[Borel91] Borel, A., Linear algebraic groups. Second edition, Springer, 1991.Google Scholar
[Lang65] Lang, S., Algebra. Addison-Wesley, 1965.Google Scholar
[Nishimura55] Nishimura, H., Some remarks on rational points. Mem. Coll. Sci. Univ. Kyoto 29(1955), 189–192.Google Scholar