Equidimensional actions of algebraic tori
Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 681-705.

Soit X une variété affine conique factorielle sur un corps algébriquement clos de caractéristique zéro. Nous considérons les actions équidimensionnelles, algébriques, et stables d’un tore algébrique sur X qui sont compatibles avec la structure conique. Nous montrons que de telles actions sont colibres et que les nilcônes de X qui lui sont associés sont des intersections complètes.

Let X be an affine conical factorial variety over an algebraically closed field of characteristic zero. We consider equidimensional and stable algebraic actions of an algebraic torus on X compatible with the conical structure. We show that such actions are cofree and the nullcones of X associated with them are complete intersections.

@article{AIF_1995__45_3_681_0,
     author = {Nakajima, Haruhisa},
     title = {Equidimensional actions of algebraic tori},
     journal = {Annales de l'Institut Fourier},
     pages = {681--705},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     doi = {10.5802/aif.1470},
     mrnumber = {96e:14055},
     zbl = {0823.14035},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1470/}
}
TY  - JOUR
AU  - Nakajima, Haruhisa
TI  - Equidimensional actions of algebraic tori
JO  - Annales de l'Institut Fourier
PY  - 1995
SP  - 681
EP  - 705
VL  - 45
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1470/
DO  - 10.5802/aif.1470
LA  - en
ID  - AIF_1995__45_3_681_0
ER  - 
%0 Journal Article
%A Nakajima, Haruhisa
%T Equidimensional actions of algebraic tori
%J Annales de l'Institut Fourier
%D 1995
%P 681-705
%V 45
%N 3
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1470/
%R 10.5802/aif.1470
%G en
%F AIF_1995__45_3_681_0
Nakajima, Haruhisa. Equidimensional actions of algebraic tori. Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 681-705. doi : 10.5802/aif.1470. http://archive.numdam.org/articles/10.5802/aif.1470/

[BK] W. Borho, H. Kraft, Über Bahenen und deren Deformationen bei linearen Aktionen reductiver Gruppen, Comment. Math. Helvetici, 54 (1979), 1-104. | EuDML | MR | Zbl

[CM] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge Studies Advanced Math., 37, Cambridge, Cambridge Univ. 1993. | MR | Zbl

[GM] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspecte der Mathematik, D1, Braunschweig-Wiesbaded, Vieweg, 1984. | MR | Zbl

[H] W.H. Hesselink, Desingularizations of varieties of nullforms, Invent. Math., 55 (1979), 141-163. | EuDML | MR | Zbl

[HR] M. Hochster, J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math., 13 (1974), 115-175. | MR | Zbl

[K] V.G. Kac, Some remarks on nilpotent orbits, J. Algebra, 64 (1980), 190-213. | MR | Zbl

[L] D. Luna, Slices étales, Bull. Soc. Math. France Mémoire, 33 (1973), 81-105. | EuDML | Numdam | MR | Zbl

[LR] M. Nagata, Local Rings, Interscience Tracts in Pure & Applied Math., 13, New York, Wiley, 1962. | Zbl

[M] A.R. Magid, Finite generation of class groups of rings of invariants, Proc. Amer. Math. Soc., 60 (1976), 45-48. | MR | Zbl

[N1] H. Nakajima, Relative invariants of finite groups, J. Algebra, 79 (1982), 218-234. | MR | Zbl

[N2] H. Nakajima, Class groups of localities of rings of invariants of reductive algebraic groups, Math. Zeit., 182 (1983), 1-15. | MR | Zbl

[N3] H. Nakajima, Representations of a reductive algebraic group whose algebras of invariants are complete intersections, J. reine angew. Math., 367 (1986), 115-138. | MR | Zbl

[N4] H. Nakajima, Equidimensional toric extensions of symplectic groups, Proc. Japan Acad., 70 Ser. A (1994), 74-79. | MR | Zbl

[N5] H. Nakajima, Semisimple algebraic groups admitting equidimensional toric extensions, in preparation.

[P1] V.L. Popov, Representations with a free module of covariants, Func. Anal. Appl., 10 (1976), 242-244. | MR | Zbl

[P2] V.L. Popov, Modern developments in invariant theory, Proc. of International Congress of Mathematicians (Berkeley 1986) Vol. 1, 394-406, Providence, Amer. Math. Soc., 1987. | Zbl

[P3] V.L. Popov, Groups, Generators, Syzygies, and Orbits in Invariant Theory, Transl. Math. Monographs 100, Providence, Amer. Math. Soc., 1992. | MR | Zbl

[S] G.W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Etudes Sci. Publ. Math., 51 (1980), 37-136. | Numdam | MR | Zbl

[TE] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Math., 339, Berlin Heidelberg New York, Springer, 1973. | MR | Zbl

[W1] D. Wehlau, A proof of the Popov conjecture for tori, Proc. of Amer. Math. Soc., 114 (1992), 839-845. | MR | Zbl

[W2] D. Wehlau, Equidimensional varieties and associated cones, J. Algebra, 159 (1993), 47-53. | MR | Zbl

Cité par Sources :