Timeline for Singularities of space curves: Open question lists?
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| Sep 12, 2021 at 7:29 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Sep 21, 2010 at 3:29 | comment | added | jlk | Here I am working over the complex numbers $\mathbb{C}$. I just noticed you are also interested in real singularities. | |
| Sep 21, 2010 at 3:28 | comment | added | jlk | One reference is Introduction to singularities and deformations by Greuel, Lossen, and Shustin. The conjecture is stated as Conjecture 1.11 on page 236. I don't think they were the first ones to state the conjecture, but I can't think of another reference off-hand. | |
| Sep 21, 2010 at 3:27 | comment | added | jlk | @Richard Montgomery: Thanks for the response. Here is an (imprecise) explanation of terminology. "Isolated" means that singular locus of the curve is $0$-dimensional, so one excludes things like $z^2=y=0$. If you are coming from the Arnol'd-Mather-Thom world, then I would guess you are already assuming this. "Rigid" means any small perturbation of the singularity is analytically equivalent to the trivial perturbation of the singularity. For example, the node $x y =0$ (a simple singularity) is not rigid because it admits the perturbation $x y= t$. | |
| Sep 20, 2010 at 20:01 | comment | added | Richard Montgomery |
Could you tell me what you mean by rigid' and isolated', or give me a reference? I come from the Arnol'd-Mather-Thom world of singularity theory, where rigid' would mean, roughly, that any small perturbation of the curve (perhaps with finite jet restrictions imposed) is locally diffeomorphic to the curve. Rigid' would then imply `simple' a la Arnol'd. (Simple unibranched space curve singularities, and so it would seem, rigid ones have been classified by Bruce and Gaffney. There are many.)
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| Sep 13, 2010 at 6:42 | history | edited | jlk | CC BY-SA 2.5 |
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| Sep 13, 2010 at 3:55 | history | answered | jlk | CC BY-SA 2.5 |