Timeline for Smoothable $\mathbb{F}_p$-variety embeds in a regular scheme
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 5, 2020 at 20:40 | vote | accept | CommunityBot | ||
| Aug 2, 2020 at 12:45 | answer | added | Jason Starr | timeline score: 0 | |
| Jul 31, 2020 at 14:15 | history | edited | user158636 | CC BY-SA 4.0 |
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| Jul 31, 2020 at 13:26 | comment | added | Jason Starr | Actually the hypothesis beginning, "Assume that ...", implies that a reduced, pure-dimensional, finite type $k$-scheme is a local complete intersection. | |
| Jul 31, 2020 at 12:15 | comment | added | user158636 | Edited in response to the comments. | |
| Jul 31, 2020 at 12:14 | history | edited | user158636 | CC BY-SA 4.0 |
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| Jul 31, 2020 at 0:59 | comment | added | Sándor Kovács | Another way this can fail is if $X$ (or equivalently the special fiber) is not Gorenstein. | |
| Jul 30, 2020 at 22:39 | comment | added | Jason Starr | I guess that the simplest explicit example is a curve $X$ in $\mathbb{P}^3$ that is the intersection of a quadric surface with a single ordinary double point $p$ and a cubic surface that has an ordinary double point at $p$ (and whose tangent cone is different from the first quadric surface). This curve has arithmetic genus $4$ and geometric genus $0$. It generizes to a smooth, canonically embedded curve of genus $4$ in $\mathbb{P}^3$. Yet the embedding dimension of the curve at $p$ equals $3$. | |
| Jul 30, 2020 at 21:19 | comment | added | Jason Starr | That is not true. If there is such a regular proper flat model, then the embedding dimension of $X$ is everywhere locally bounded by $1+\text{dim}(X)$. Now consider a reduced curve with embedding dimension $3$ and apply the Hilbert-Burch(-Schaps) theorem. | |
| Jul 30, 2020 at 19:12 | history | asked | user158636 | CC BY-SA 4.0 |