Timeline for Kummer theory and ramified covers
Current License: CC BY-SA 3.0
12 events
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| Jun 24, 2015 at 9:23 | vote | accept | zyx | ||
| Jun 24, 2015 at 5:58 | comment | added | Francesco Polizzi | You can use the same procedure, cant'you? Call $B$ your set of points and let $f$ be a generator of the extension. If $f$ does not vanish at $B$ you are done; otherwise, replace $f$ with $f+\pi^*g$, where $g \in K(Y)$ is any rational function non vanishing at the points of the set $\pi(B)$. In fact, if $b \in B$ we have $$(f+\pi^*g)(p) = f(p) + g(\pi(b)) = g(\pi(b)) \neq 0.$$ | |
| Jun 24, 2015 at 3:48 | comment | added | zyx | Sorry, I think the question was not clear. I fix a set of points outside the ramification locus of the morphism. Is it possible to find a generator $f$ not vanishing at these points? | |
| Jun 23, 2015 at 18:51 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jun 23, 2015 at 18:50 | comment | added | Francesco Polizzi | Assume that your generator $f$ vanishes only at the ramification points of $\pi \colon X \to Y$. If $g \in K(Y)$ is such that $g$ does not vanish on the branch locus of $\pi$, then $f + \pi^*g$ is a generator of the extension that does not vanish on the ramification locus of $\pi$. | |
| Jun 23, 2015 at 14:33 | comment | added | zyx | I guess this should be possible modifying with functions in $K(Y)$, am I right? | |
| Jun 21, 2015 at 13:32 | vote | accept | zyx | ||
| Jun 21, 2015 at 21:21 | |||||
| Jun 21, 2015 at 13:32 | comment | added | zyx | Thanks for your answer, Francesco. Of course, you are right. But, if I a finite set of points is fixed, is it possible to find $f$ not having zeros nor poles on these points? | |
| Jun 21, 2015 at 12:42 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jun 21, 2015 at 12:35 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jun 21, 2015 at 10:13 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Jun 21, 2015 at 8:50 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |