Abstract
These brief notes have their origin in a conversation I had with Abhyankar during a pleasant walk along the Wabash a few years ago. While I tried to explain some ideas on torus actions on affine spaces Abhyankar, in that inimitable way he has to concentrate one’s mind, feigned innocence as to such fancy notions, but was quite happy to listen when I proposed talking about gradings of polynomial rings instead. The two topics cover exactly the same ground from two different points of view, equally valid and valuable. Torus actions also provide an uncomplicated introduction to the larger subject of reductive group actions. I gave my conference talk hoping that such an introduction in the readily understood language of gradings would be appreciated by some of the non-experts in the rather diverse audience Abhyankar’s birthday was bound to bring together. This is a slightly expanded version of my talk. It is entirely expository in nature and not meant for the specialists.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abhyankar, S. S. and Moh, T. T., Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166.
Bass, H. and Haboush, W., Linearizing certain reductive group actions, Trans. Am. Math. Soc. Vol.292, No.2, (1985), 463–481.
Bialynicki-Birula, A., Remarks on the action of an algebraic torus on kn, I and II, Bull. Acad. Pol. Sci., 14 (1966), 177–181 and 15 (1967), 123–125.
Floyd, E. E., On periodic maps and the Euler characteristic of associated spaces, Trans. Am. Math. Soc. 72 (1952), 138–147.
Fogarty, J., Fixed point schemes, Amer. J. Math. 95 (1973), 35–51.
Fujita, T., On Zariski problem, Proc. Japan Acad., 55A (1979), 106–110.
Kambayashi, T. and Russell, P., On linearizing algebraic torus actions, J. Pure and Applied Algebra 23 (1982), 243–250.
Koras, M., A characterization of C 2/Z a, to appear.
Koras, M. and Russell, P., Gm-actions on A3, Canadian Math. Soc. Conference Proceedings, Vol. 6 (1986), 269–276.
Koras, M. and Russell, P., On linearizing “good” C*-actions on C 3, Canadian Math. Soc. Conference Proceedings, Vol.10 (1989), 93–101.
Koras, M. and Russell, P., Codimension 2 torus actions on affine n-space, Canadian Math. Soc. Conference Proceedings, Vol. 10 (1989), 103–110.
Koras, M. and Russell, P., Mixed C*-actions on C 3 with isolated fix-point, in preparation.
Kraft, H. and Schwarz, G., Reductive group actions on affine space with one-dimensional quotient, Canadian Math. Soc. Conference Proceedings, Vol.10 (1989), 125–132.
Kraft, H., Petrie, T. and Randall, J., Quotient varieties, Adv. Math. 74, No. 2(1989), 145–162.
Miyanishi, M. and Sugie, T., Affine surface containing cylinderlike open sets, J. Math. Kyoto Univ. 20 (1980), 11–42.
Quillen, D., Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171.
Russell, P., On affine-ruled rational surfaces, Math. Ann. 255 (1981), 287–302.
Suslin, A. A., On projective modules over polynomial rings, Math. USSR Sbornik 22 (1974) 595–602.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Russell, P. (1994). Gradings of Polynomial Rings. In: Bajaj, C.L. (eds) Algebraic Geometry and its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2628-4_22
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2628-4_22
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7614-2
Online ISBN: 978-1-4612-2628-4
eBook Packages: Springer Book Archive