Timeline for A question on smoothness of varieties
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 4, 2012 at 15:48 | vote | accept | gio | ||
| Feb 4, 2012 at 6:09 | answer | added | Sándor Kovács | timeline score: 2 | |
| Feb 4, 2012 at 0:48 | comment | added | Charles Staats | (Incidentally, under your other hypotheses, the dimension hypothesis is equivalent to the statement that $\mathscr{O}_{X,x}$ is flat over the local ring at $\pi(x)$.) | |
| Feb 4, 2012 at 0:46 | comment | added | Charles Staats | There may be an obvious argument that the dimension hypothesis always holds or an obvious counterexample, but I'm not seeing either at the moment. | |
| Feb 4, 2012 at 0:44 | comment | added | Charles Staats | I'm fairly sure the answer is yes IF $\operatorname{dim} X = \operatorname{dim} \overline{\pi_L(X)} + \operatorname{dim} \overline{\pi_L^{-1}(\pi_L(x))}$. Idea: take a regular sequence that locally generates the maximal ideal $\mathfrak{m}$ of $\pi(x)$, and pull it back it to $\mathscr{O}_{X,x}$. By standard properties of Cohen-Macaulay rings, the pullback sequence must still be exact; it also generates $\mathfrak{m}\mathscr{O}_{X,x}$. Smoothness of the fiber then allows us to extend the regular sequence to a regular sequence generating the maximal ideal of $\mathscr{O}_{X,x}$. | |
| Feb 3, 2012 at 19:22 | history | edited | gio | CC BY-SA 3.0 |
deleted 5 characters in body
|
| Feb 3, 2012 at 18:55 | comment | added | Charles Staats | Will: smooth schemes over a perfect field (and probably much more generally, but I want to speak carefully) are always reduced, since a regular local ring is always a UFD and, in particular, a domain. | |
| Feb 3, 2012 at 18:47 | comment | added | Will Sawin | Is $\pi_L^{-1}$ taking the fiber? Because in that case, the projection of the nodal cubic $y^2=x^2(x+1)$ or the cuspoidal curve $y^2=x^3$ onto the $x$ axis is just the whole line, smooth, and the fiber at the origin is just the double point, presumably smooth. | |
| Feb 3, 2012 at 17:27 | history | edited | agt |
edited tags
|
|
| Feb 3, 2012 at 17:22 | history | asked | gio | CC BY-SA 3.0 |