Timeline for Does the link of a hypersurface singularity determine its analytic type?
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6 events
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| Oct 21, 2020 at 22:37 | comment | added | Jason Starr | The example in my previous comment is wrong, but Eliashberg-Mishachev does yield a counterexample. I will try to post it as an answer soon . . . | |
| Oct 15, 2020 at 9:41 | comment | added | Jason Starr | I think that the isotopy class (maybe even with some extra structure coming from the contact manifold structure on the sphere) does not determine the deformation class. Consider the blowings up of projective 3-space at both a twisted cubic (rational) curve and a plane cubic (elliptic) curve. Choose a projective embedding of each having the same degree on the strict transform of a general line and on a (rational curve) fiber of the exceptional divisor over the center of the blowing up. In the h-principle book by Eliashberg and Mishachev they prove a symplectic isotopy theorem . . . | |
| Oct 8, 2020 at 23:44 | comment | added | dorebell | I meant something more like "up to isotopy" than "up to diffeomorphism"; e.g. in the case $n = 2$, $L$ is an actual link in $S^3$, and I certainly want to remember more than the number of connected components. (I have a vague recollection of seeing a more canonical definition of the link, avoiding the dependence on $\epsilon$). Your counterexample then shows that the most we could hope for is that the isotopy class of the link determines the deformation-equivalence class of the singularity. | |
| Oct 2, 2020 at 8:45 | comment | added | Jason Starr | What do you mean when you write, "In particular, does $L$, specified as a submanifold of $S^{2n+1}_\epsilon$, completely determine the analytic type of the singularity?" Since these submanifolds of spheres vary with $\epsilon$, I assume that you mean "up to diffeomorphism". But now consider a nonisotrivial family of smooth hypersurfaces in $\mathbb{CP}^n$ and the cones over these. | |
| May 4, 2016 at 17:09 | comment | added | roy smith | do you know Mumford's thesis? maths.ed.ac.uk/~aar/papers/mumfordsin.pdf | |
| May 4, 2016 at 7:08 | history | asked | dorebell | CC BY-SA 3.0 |