The logarithmic limit-set of an algebraic variety
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- by George M. Bergman PDF
- Trans. Amer. Math. Soc. 157 (1971), 459-469 Request permission
Abstract:
Let $C$ be the field of complex numbers and $V$ a subvariety of ${(C - \{ 0\} )^n}$. To study the “exponential behavior of $V$ at infinity", we define $V_\infty ^{(a)}$ as the set of limitpoints on the unit sphere ${S^{n - 1}}$ of the set of real $n$-tuples $({u_x}\log |{x_1}|, \ldots ,{u_x}\log |{x_n}|)$, where $x \in V$ and ${u_x} = {(1 + \Sigma {(\log |{x_i}|)^2})^{ - 1/2}}$. More algebraically, in the case of arbitrary base-field $k$ we can look at places “at infinity” on $V$ and use the values of the associated valuations on ${X_1}, \ldots ,{X_n}$ to construct an analogous set $V_\infty ^{(b)}$. Thirdly, simply by studying the terms occurring in elements of the ideal $I$ defining $V$, we define another closely related set, $V_\infty ^{(c)}$. These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of $GL(n,Z)$ on $k[X_1^{ \pm 1}, \ldots ,X_n^{ \pm 1}]$, then studied further. It is shown among other things that $V_\infty ^{(b)} = V_\infty ^{(c)} \supseteq$ (when defined) $V_\infty ^{(a)}$. If a certain natural conjecture is true, then equality holds where we wrote “$\supseteq$", and the common set ${V_\infty } \subseteq {S^{n - 1}}$ is a finite union of convex spherical polytopes.References
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- George M. Bergman, A weak Nullstellensatz for valuations, Proc. Amer. Math. Soc. 28 (1971), 32–38. MR 272780, DOI 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.
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 157 (1971), 459-469
- MSC: Primary 14.01
- DOI: https://doi.org/10.1090/S0002-9947-1971-0280489-8
- MathSciNet review: 0280489