Timeline for Properties of divisors when moving from char 0 to char p.
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 21, 2013 at 3:06 | vote | accept | user32134 | ||
| Aug 15, 2013 at 19:20 | history | edited | Jason Starr | CC BY-SA 3.0 |
added 57 characters in body
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| Aug 15, 2013 at 19:17 | comment | added | Jason Starr | @Daniel: "Open" should be "stable under generization". Let $(X_R,\mathcal{O}(1))$ be a projective, flat $R$-scheme, where $R$ is a DVR. Let $\mathcal{L}_R$ be an invertible sheaf on $X_R$. Assume that $\mathcal{L}_0$ is nef on the closed fiber $X_0$. Then for every positive integer $N$, $\mathcal{L}_0^{\otimes N}(1)$ is ample on $X_0$ by Kleiman's criterion. Thus, by openness of ampleness, $\mathcal{L}_R^{\otimes N}(1)$ is ample for every positive integer $N$. Thus, $\mathcal{L}_R$ is nef. | |
| Aug 15, 2013 at 16:52 | comment | added | Daniel Litt | I should remark, the example I linked to is for an $\mathbb{R}$-divisor (theorem 1.2); I don't know of an honest example for a Cartier divisor, but I don't see why nefness should be open in that case either. | |
| Aug 15, 2013 at 16:37 | comment | added | Daniel Litt | I'm not sure quite what you mean by your parenthetical remark in the first line, but nefness is not an open property--see e.g. math.mit.edu/~johnl/docs/bminus.pdf | |
| S Aug 15, 2013 at 13:56 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Aug 15, 2013 at 13:56 | history | made wiki | Post Made Community Wiki by Jason Starr |