Timeline for Viewing an algebraic subset through hyperplane sections
Current License: CC BY-SA 4.0
13 events
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| Jan 14, 2022 at 19:44 | history | edited | Zerox | CC BY-SA 4.0 |
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| Jan 12, 2022 at 12:09 | history | edited | Zerox | CC BY-SA 4.0 |
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| Jan 12, 2022 at 12:00 | history | edited | Zerox | CC BY-SA 4.0 |
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| Jan 12, 2022 at 11:56 | comment | added | Zerox | @JasonStarr How about $n>2$ case? | |
| Jan 12, 2022 at 11:55 | comment | added | Jason Starr | Then the result is wrong. Start with a plane curve and remove a subset. | |
| Jan 12, 2022 at 11:53 | comment | added | Zerox | @JasonStarr I'm not saying "$H$ intersects $X$ with multiplicity". I mean the intersection $H \cap X$ can be seen as a degree $k$ hypersurface in $H$. In $n=2$ case, this means at most $k$ points. I'm changing the question to deal with this case. | |
| Jan 12, 2022 at 11:50 | comment | added | Jason Starr | For an arbitrary subset of $\mathbb{CP}^2$, how are you defining "intersection multiplicity" for the intersection with a line at a point? I do not understand your hypothesis for subsets of $\mathbb{CP}^2$. | |
| Jan 12, 2022 at 11:42 | history | edited | Zerox | CC BY-SA 4.0 |
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| Jan 12, 2022 at 11:35 | comment | added | Zerox | @AntoineLabelle Just as Francesco Polizzi said, a degree $k$ "hypersurface" in $\Bbb{CP}^1$ is the zero locus of a single degree $k$ polynomial, which is $k$ points with multiplicity. | |
| Jan 12, 2022 at 6:32 | comment | added | Francesco Polizzi | For $n=2$, the OP is requiring that the intersection with every line is a fixed number of points (possibly with multiplicity), say $k$. Is there a $X$ with this property that is not an algebraic curve of degree $k$? | |
| Jan 12, 2022 at 3:45 | comment | added | Antoine Labelle | For $n=2$, wouldn't any subset $X$ that intersects any line in finitely many points (but at least one) be a counterexample? There certainly exist such subsets that are not algebraic hypersurfaces. | |
| Jan 11, 2022 at 14:17 | history | edited | Zerox | CC BY-SA 4.0 |
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| Jan 11, 2022 at 14:11 | history | asked | Zerox | CC BY-SA 4.0 |