The logarithmic limit-set of an algebraic variety
George M. Bergman
Trans. Amer. Math. Soc. 157 (1971), 459-469
Full-text PDF Free Access
Similar Articles |
Abstract: Let be the field of complex numbers and a subvariety of . To study the ``exponential behavior of at infinity", we define as the set of limitpoints on the unit sphere of the set of real -tuples , where and . More algebraically, in the case of arbitrary base-field we can look at places ``at infinity'' on and use the values of the associated valuations on to construct an analogous set . Thirdly, simply by studying the terms occurring in elements of the ideal defining , we define another closely related set, .
These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of on , then studied further.
It is shown among other things that (when defined) . If a certain natural conjecture is true, then equality holds where we wrote ``", and the common set is a finite union of convex spherical polytopes.
Dnestrovskaya tetrad: Nereshennye zadachi teorii kolets i
modulei, First All-Union Symposium on the Theory of Rings and Modules
(Kishinev, 1968), Redakc.-Izdat. Otdel Akad. Nauk Moldav. SSR, Kishinev,
1969 (Russian). MR 0254084
M. Bergman, A weak Nullstellensatz for
valuations, Proc. Amer. Math. Soc. 28 (1971), 32–38. MR 0272780
(42 #7661), http://dx.doi.org/10.1090/S0002-9939-1971-0272780-1
D. Mumford, Introduction to algebraic geometry, Department of Mathematics, Harvard University, Cambridge, Mass., 1966 (preliminary version of first three chapters).
M. Raynaud, Modèles de Néron, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A345-A347. MR 33 #2631.
Zariski and Pierre
Samuel, Commutative algebra. Vol. II, The University Series in
Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.
J.-Toronto-London-New York, 1960. MR 0120249
- Dniestr Notebook, unsolved problems in the theory of rings and modules, L. M. Andreichuk, Editor, 500 copies duplicated by the Acad. Sci. Moldavian SSR, June 1969, 6 kopeks. (Collection of 101 problems from the Kishinyev symposium on ring and module theory, 3-6 September, 1968. All but one of the other problems are for noncommutative or nonassociative rings. Russian.) MR 0254084 (40:7294)
- G. M. Bergman, A weak Nullstellensatz for valuations, Proc. Amer. Math. Soc. 28 (1971), 32-38. MR 0272780 (42:7661)
- D. Mumford, Introduction to algebraic geometry, Department of Mathematics, Harvard University, Cambridge, Mass., 1966 (preliminary version of first three chapters).
- M. Raynaud, Modèles de Néron, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A345-A347. MR 33 #2631.
- O. Zariski and P. Samuel, Commutative algebra. Vol. 2, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)
Retrieve articles in Transactions of the American Mathematical Society
Retrieve articles in all journals
exponential behavior at infinity,
general linear group,
sphere at infinity
© Copyright 1971 American Mathematical Society