Timeline for Finiteness of geometric valuations
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 31, 2015 at 18:07 | vote | accept | Li Yutong | ||
| Mar 31, 2015 at 16:26 | answer | added | Karl Schwede | timeline score: 3 | |
| Mar 31, 2015 at 14:27 | comment | added | Karl Schwede | @Mark If you include the KLT assumption, you actually get the finiteness of $E$ for any set of of divisors with log discrepancy $\leq n$ (for any $n$). | |
| Mar 31, 2015 at 14:25 | comment | added | Karl Schwede | @LiYutong Unfortunately, the set is still not finite. You need KLT singularities. If you have two exceptional divisors over a surface $X$, say $E_1, E_2$ with discrepancy $-1$, then blowing up the intersection point will result in a third divisor with discrepancy $-1$, say $E_3$. If you blow up the intersection of $E_3$ with $E_1$ you get a fourth divisor, etc. | |
| Mar 31, 2015 at 14:25 | comment | added | user47305 | You really want log discrepancy $\leq 1$ (log discrepancy is discrepancy + 1). There are only finitely many because all such divisors must be extracted on any resolution of $X$. | |
| Mar 31, 2015 at 13:58 | comment | added | Li Yutong | Dear@KarlSchwede, I had edited the condition: I do mean a variety $X$ with reasonable singularities, and the exceptional divisor with discrepancy $\leq 1$. Why in this case such $E$ is finite? | |
| Mar 31, 2015 at 13:57 | history | edited | Li Yutong | CC BY-SA 3.0 |
added 43 characters in body
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| Mar 31, 2015 at 4:02 | comment | added | Karl Schwede | Will Sawin is certainly right, there are infinitely many such $E$. But if you put stronger conditions on the singularities of $X$ (like log terminal and $\mathbb{Q}$-Gorenstein) and stronger conditions on the $E$ (like log discrepancy $\leq 1$) then the set is indeed finite... Or do you mean to consider $E$ that appear on every resolution (not just some resolution?) | |
| Mar 31, 2015 at 3:24 | comment | added | Will Sawin | But those aren't finite either, as iterated blowing up of a point in $\mathbb P^2$ shows. | |
| Mar 31, 2015 at 2:06 | comment | added | Allen Knutson | Since these $E$ aren't in $X$, but in various possibilities for $Y$, I'd say there's a proper class of such $E$. Are you really looking for something like the set of valuations on the structure sheaf of $X$, centered at $Z$? | |
| Mar 31, 2015 at 1:39 | history | asked | Li Yutong | CC BY-SA 3.0 |