Timeline for Movable Divisors
Current License: CC BY-SA 3.0
8 events
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| Jan 5, 2015 at 5:33 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 5, 2015 at 5:26 | comment | added | Sándor Kovács | Mark, you are absolutely correct. Kawamata defines a movable divisor as you suggested. Unfortunately, it still seems to be different from the definition of Lazarsfeld (and others) of a movable curve. So on a surface a curve could be movable as a divisor, but not as a curve. :( | |
| Jan 5, 2015 at 5:19 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 5, 2015 at 5:14 | comment | added | Sándor Kovács | Mark, thanks for the explanation. This makes sense. I had actually looked at Matsuki's book. He defines the movable cone indeed (which I forgot by the time I wrote the answer), but I couldn't find his definition of a movable divisor... Anyway, what you are saying makes perfect sense and indeed, for CYs, or more generally for flips and flops this notion is invariant. I'm convinced. Thanks. | |
| Jan 5, 2015 at 5:00 | comment | added | user47305 | The notation of $\text{Mov}(X)$ for a cone of divisors, while perhaps unfortunate, is not so uncommon. Off the top of my head, you can find it in the Hu-Keel MDS paper, Matsuki's intro MMP book, or several papers on the Kawamata-Morrison cone conjecture (which is in part a conjecture on the structure of the movable cone of divisors on a CY). | |
| Jan 5, 2015 at 4:59 | comment | added | user47305 | Your answer seems right to me -- nice! Just a couple remarks on "movable divisor" -- I think the notion of having small base locus at least deserves a name. For example, if X is Calabi-Yau (or MDS), then Mov(X) is (conjecturally) the union of the strict transforms of the nef cones of all the small modifications. It's true, as you say, that this isn't closed under pullbacks, but it is closed under pushforward via any map that extract no divisors (for example, an arbitrary run of the MMP, where we might want our $\Delta$ to lie in this class). | |
| Jan 5, 2015 at 1:58 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Jan 5, 2015 at 1:10 | history | answered | Sándor Kovács | CC BY-SA 3.0 |