Timeline for Do proper polynomial mappings have a path-lifting property?
Current License: CC BY-SA 2.5
8 events
when toggle format | what | by | license | comment | |
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Nov 23, 2010 at 19:40 | vote | accept | Kevin M Pilgrim | ||
Nov 23, 2010 at 19:27 | history | edited | Sergei Ivanov | CC BY-SA 2.5 |
fixed tupos
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Nov 23, 2010 at 19:22 | comment | added | BCnrd | Dear Sergei: Shouldn't $\delta$ and $\varepsilon$ also be swapped at the end of the first sentence of (5)? (That is, for $S$ of diameter at least $\varepsilon$, $f(S)$ has diameter at least $\delta$. We seek a uniform positive lower bound on the diameter of $f(S)$, given such a lower bound on the diameter of $S$.) | |
Nov 23, 2010 at 18:55 | comment | added | Sergei Ivanov | @BCnrd: thanks, renamed $\delta_0$ to $\varepsilon_0$. For finiteness of pre-images, I had in mind the following argument: the first coordinates of roots of $f$ are roots of some polynomial $f_1:\mathbb C\to\mathbb C$ (that can be written explicitly via resultants or whatever). If this polynomial is zero, then the set of roots of $f$ is non-compact and hence non-proper. Otherwise the number of possible values of first coordinates of roots of $f$ is bounded by the degree of $f_1$ which is bounded in terms of the degree of $f$. Applying this argument to other coordinates yields the result. | |
Nov 23, 2010 at 18:46 | history | edited | Sergei Ivanov | CC BY-SA 2.5 |
clarified notation
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Nov 23, 2010 at 17:48 | comment | added | BCnrd | Dear Buschi: Your example $f$ is not proper. Sergei is correct about (1) because a proper affine morphism has finite fibers (and is even a finite morphism) and a map between separated finite type $\mathbf{C}$-schemes is proper if and only if the map on $\mathbf{C}$-points is topologically proper (for the analytic topology). There is a small typo in the argument, however: near the end of the first sentence of (5) the roles of $\varepsilon$ and $\delta$ should be swapped (and later in (5) the role of $\delta_0$ should be called $\varepsilon_0$, though logically it is OK to write $\delta_0$). | |
Nov 23, 2010 at 14:14 | comment | added | Buschi Sergio | You write: (1) The pre-image of every point if finite... but let $g: C^n\to C$ polynomial, and $f:=(g,..,g): C^n\to C^n$ then $f^{-1}(z,...,z)=g^{-1}(z)$ is a (n-1)-variety. | |
Nov 23, 2010 at 8:06 | history | answered | Sergei Ivanov | CC BY-SA 2.5 |