Timeline for reference for "curves over S are locally the base change of a curve over S' which is finite type over R"
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 25, 2015 at 19:59 | vote | accept | Will Chen | ||
| Feb 24, 2015 at 9:07 | answer | added | Laurent Moret-Bailly | timeline score: 7 | |
| Feb 24, 2015 at 8:01 | comment | added | Laurent Moret-Bailly | More trivially, you can take for $S$ a disjoint union of points $s_n=\mathrm{Spec}(k_n)$ ($n\in\mathbb{N}$ and $k_n$ a field) and $X=\coprod_n X_n$ where $X_n$ is smooth projective of genus $n$ over $k_n$. | |
| Feb 23, 2015 at 22:53 | comment | added | Jason Starr | "... what's your counterexample?" Let $A$ be $\mathbb{Q}[x_1,x_2,x_3,\dots]$, let $\mathfrak \subset A$ be $\langle x_1,x_3,\dots \rangle$. Let $S$ be $\text{Spec}(A)\setminus \{ \mathfrak{m} \}$. This is the union of the countably many open subsets $U_n = D(x_1)\cup \dots \cup D(x_n)$. Over each open subset $U_n$, $X\times_S U_n$ will be the blowing up of $\mathbb{P}^1\times U_n$ along a particular closed subscheme. For $U_2$, blowup $s(U_2) = \{[1,0]\}\times U_2$ over $Z(x_1)$. Then, over $U_3$, blowup further the strict transform of $s(U_3)$ over $Z(x_1,x_2)$, etc. | |
| Feb 23, 2015 at 22:29 | comment | added | Will Chen | @JasonStarr I wasn't, what's your counterexample? | |
| Feb 23, 2015 at 22:15 | comment | added | Jason Starr | Are you assuming that $S$ is quasi-compact? I think you can make a counterexample if $S$ is not quasi-compact. | |
| Feb 23, 2015 at 22:01 | history | asked | Will Chen | CC BY-SA 3.0 |