Timeline for Comparisons of log canonical thresholds
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2018 at 16:19 | comment | added | Maurizio Moreschi | I meant components of multiplicity bigger than 1. | |
| Oct 30, 2018 at 16:12 | comment | added | Zach Teitler | $E$ might have multiple components, they are just pairwise disjoint. | |
| Oct 30, 2018 at 15:45 | vote | accept | Maurizio Moreschi | ||
| Oct 30, 2018 at 15:31 | history | edited | Maurizio Moreschi | CC BY-SA 4.0 |
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| Oct 30, 2018 at 15:15 | comment | added | Maurizio Moreschi | Yes, I agree. Hironaka's construction behaves well under extensions of the base fields. So after base change I still have a log-resolution. Also, if a divisor E is smooth and irreducible, then after base change might not be irreducible, but it is still smooth, so it cannot have multiple components. It follows that the numerical data are the same (apart possibly being repeated more times). | |
| Oct 30, 2018 at 14:22 | comment | added | Karl Schwede | I agree with Zach, lct over ${\mathbb Q}$ and over ${\mathbb C}$ are the same for the above reason. Explicitly, if I have a log resolution of $({\mathbb Q}, f)$ where $X$ and $f$ are defined over ${\mathbb Q}$, then after base change, it is still a log resolution over ${\mathbb C}$. | |
| Aug 31, 2018 at 17:32 | comment | added | Zach Teitler | Hmmm, I wonder if it is perhaps simpler than this? Start with a log resolution over the smaller field $K$. It remains a log resolution over $L$, does it not? The only change is that possibly some of the exceptional divisors which were irreducible over $K$ become reducible over $L$, but in that case all the multiplicities stay the same. You just have several copies (a Galois orbit) of "the same" exceptional divisor. One must be careful, but if the original singularity was defined over $K$ then I think the $K$-resolution is still an $L$-resolution...? | |
| Aug 1, 2018 at 10:12 | history | edited | Maurizio Moreschi | CC BY-SA 4.0 |
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| Aug 1, 2018 at 10:07 | history | answered | Maurizio Moreschi | CC BY-SA 4.0 |