Timeline for What is the most simple non-planar Gorenstein curve singularity?
Current License: CC BY-SA 2.5
11 events
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| May 31, 2010 at 21:17 | history | edited | Graham Leuschke | CC BY-SA 2.5 |
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| May 27, 2010 at 23:40 | history | edited | Graham Leuschke | CC BY-SA 2.5 |
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| May 27, 2010 at 14:24 | comment | added | Graham Leuschke | Ah, so the delta-invariant is just the length of $R/I$, where $I$ is the conductor. Now I see. I also see that it is indeed referred to as "delta" in the first lines of Bass's Ubiquity paper (though it never appears later in the paper). OK, so this answer is garbage, but I don't think I can delete it without losing these useful comments, so I'll leave it up. | |
| May 27, 2010 at 2:29 | comment | added | jlk | @Graham, The number $\operatorname{dim} \tilde{R}/R$ is a basic invariant of the curve. The Gorenstein relation is, as you probably know, equivalent to the equation $2 \cdot \operatorname{dim} \tilde{R}/R = \operatorname{dim} \tilde{R}/I$, where $I$ is the conductor ideal. | |
| May 27, 2010 at 2:23 | history | edited | Graham Leuschke | CC BY-SA 2.5 |
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| May 26, 2010 at 23:51 | comment | added | Graham Leuschke | Hmm, that's a problem. Hailong's right. I think I confused $\tilde{R}/\mathfrak{m}$ with $\tilde{R}/\mathfrak{m}\tilde{R}$. Drat, it was such a clean and satisfying answer too. jlk, I don't know a standard term for that dimension -- it's never come up in my experience before. Why is it useful/interesting? | |
| May 26, 2010 at 22:21 | comment | added | jlk | btw, is "degree" the standard term for the dimension of \tilde{R}/R? | |
| May 26, 2010 at 22:20 | comment | added | jlk | I'm also a little confused. If we take $R$ to be the subring of the product of 4 power series rings generated by (t,0,0,-t), (0,t,0,-t), (0,0,t,-t) (4 general lines in 3-space), then \tilde{R}/R has basis given by (1,0,0,0), (0,1,0,0), (0,0,1,0), and (t,0,0,0). If m is the maximal ideal, then I am computing that m^{n}/m^{n+1} = 4 for n sufficiently large. I think this means the multiplicity is 4. Am I using the wrong definition of multiplicity or something? | |
| May 26, 2010 at 21:59 | comment | added | Hailong Dao | I am a little confused about Ingredient 1: if $R=k[[t^a,t^b]]$ with $a<b$ then the multiplicity is $a$, while the length of the quotient = the number of integers not in the semigroup generated by $(a,b)$ =$(a-1)(b-1)/2$. | |
| May 26, 2010 at 19:43 | history | edited | Graham Leuschke | CC BY-SA 2.5 |
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| May 26, 2010 at 19:34 | history | answered | Graham Leuschke | CC BY-SA 2.5 |