Timeline for Zariski closure of hypersurfaces with $k$ singularities
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 19, 2015 at 14:02 | vote | accept | Hans | ||
| Oct 19, 2015 at 12:57 | comment | added | Francesco Polizzi | Uhm, it seems to me that for an isolated singularity the length coincides with the Milnor number, since the Milnor number is defined by taking the quotient of the local ring by the Jacobian ideal, right? | |
| Oct 19, 2015 at 12:48 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
added 18 characters in body
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| Oct 19, 2015 at 12:47 | answer | added | Francesco Polizzi | timeline score: 3 | |
| Oct 19, 2015 at 11:39 | comment | added | Jason Starr | Yes, that is true. Consider the closed subscheme of a hypersurface that is defined by the Fitting ideal of the sheaf of relative differentials. The dimension of this closed subscheme is upper semicontinuous, so the parameter point $[X]\in \mathbb{P}(V)$ cannot be in the Zariski closure of the closed locus parameterizing hypersurfaces whose singular scheme has positive dimension. For nodal hypersurfaces, the length of the singular scheme is precisely the number of nodes, whereas for $k$-singular schemes, it is at least $k$. Finally, the length is upper semicontinuous. | |
| Oct 19, 2015 at 11:04 | history | asked | Hans | CC BY-SA 3.0 |