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Timeline for Open locus ordinary double points

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Aug 6, 2013 at 11:48 comment added Jason Starr @Roy and Francesco: Of course what you both say is completely true. But there is some work to show that the formal maps to the formal deformation spaces, and thus also the inverse image of the ODP locus, "algebraize". The argument below is one way to do that work. (Of course that argument only applies to ODP's, not more general classes of singularities that are stable under infinitesimal deformation.)
Aug 6, 2013 at 8:33 comment added Francesco Polizzi A normal form of a ODP is $$x_1^2+ \cdots +x_n^2=0.$$ The Tjurina algebra is $1$-dimensional, hence a semiuniversal deformation of this hypersurface singularity is $$x_1^2+ \cdots +x_n^2+tg(x_1, \ldots, x_n)=0.$$ This shows that an ODP either stay the same or smooths under deformation, as Roy said. A reference is Greuel-Lossen-Shustin Introduction to singularities and deformations, Corollary 1.17 page 239.
Aug 6, 2013 at 4:27 comment added LMN @Roy, could you give a reference where one could learn about the deformation theory of odp's as you mention?
Aug 6, 2013 at 2:30 comment added roy smith the point is that odp's can only either stay the same or smooth under deformation. this follows from the deformation theory of odp's. hence this set is open.
Aug 5, 2013 at 20:49 answer added HNuer timeline score: 1
Aug 5, 2013 at 19:20 answer added Jason Starr timeline score: 5
Aug 5, 2013 at 17:29 review First posts
Aug 5, 2013 at 17:44
Aug 5, 2013 at 17:10 history asked none CC BY-SA 3.0