Timeline for Do singularities of plane curves deform independently?

8 events

when toggle format	what		by	license	comment
Aug 17, 2016 at 15:49	vote	accept	DCT
Aug 17, 2016 at 8:39	answer	added	Francesco Polizzi		timeline score: 5
Aug 17, 2016 at 8:32	comment	added	Francesco Polizzi		@JasonStarr: I think that the original work of Severi already answers the question about the possibility of independently smoothing the singularity of nodal curves. In fact, the proof of this result only involves checking the smoothness of some deformation space, and Severi's proof was in this aspect correct (as far I can remember). Harris solved the much harder problem of proving its irreducibility, for which Severi's proof was flawed.
Aug 17, 2016 at 8:21	history	edited	Francesco Polizzi	CC BY-SA 3.0	edited title
Aug 15, 2016 at 22:42	comment	added	Jason Starr		For curves that are worse than nodal, this map is not surjective. For instance, consider the "hypercuspidal curve", $C=\text{Zero}(Y^d - X Z^{d-1})$. In this case, $\textit{Ext}^1 ( \Omega_C, \mathcal{O}_C )$ is a torsion sheaf supported at the unique singular point $[X,Y,Z] = [1,0,0]$ with length $(d-1)(d-2)$. On the other hand $k[X,Y,Z]_d$ has length $(d+2)(d+1)/2$. For $d\geq 9$, the map cannot be surjective. For the nodal case, you might check Harris's paper solving the Severi conjecture -- I remember something about this in that paper.
Aug 15, 2016 at 21:17	history	edited	DCT	CC BY-SA 3.0	added 222 characters in body
Aug 14, 2016 at 22:50	history	edited	DCT		edited tags
Aug 14, 2016 at 22:30	history	asked	DCT	CC BY-SA 3.0