When is there a deformation of a given singularity to a normal singularity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:57:36Z http://mathoverflow.net/feeds/question/86003 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86003/when-is-there-a-deformation-of-a-given-singularity-to-a-normal-singularity When is there a deformation of a given singularity to a normal singularity Karl Schwede 2012-01-18T15:11:59Z 2012-06-01T03:23:13Z <p><strong>Question:</strong> Given a variety $X_0$ with a singularity (say Cohen-Macaulay), when does this exist as a special fiber of a flat family $X \to C$ mapping to a smooth curve $C$, such that the generic fiber is normal? </p> <p>My one thought is that perhaps this reduces to checking whether some 1-dimensional thing is smoothable via some Bertini argument perhaps (using the R1 + S2 characterization of normality).</p> <p>Of course, maybe there is some easy reference on deformations of singularities which answers this? (Hopefully)</p> <p><strong>Example:</strong> I actually have a very specific example in mind whose presentation is given below:</p> <p>$$\frac{\mathbb{F}_2[[a,b,c,d]]}{(bc, d^2 + ab^2, dc)}.$$</p> <p>Feel free to take the algebraic closure of the field by the way. I can also discuss how it comes about if it is relevant and I can also describe how one can do an integral domain version if it helps.</p> <p>This example is Cohen-Macaulay, has two components, and is 2-dimensional (checked via Macaulay2), so I can't use positive characteristic analogs of the sort of results for Du Bois singularities saying that various non-CM things can't be deformed to Cohen-Macaulay things as in Kollar-Kovacs (such theorems do hold in positive characteristic but are not published).</p> <p>This example is F-injective (the characteristic 2 analog of Du Bois) but it is not F-pure (the characteristic 2 analog of semi log canonical).</p> http://mathoverflow.net/questions/86003/when-is-there-a-deformation-of-a-given-singularity-to-a-normal-singularity/86134#86134 Answer by inkspot for When is there a deformation of a given singularity to a normal singularity inkspot 2012-01-19T19:55:59Z 2012-01-19T19:55:59Z <p>Your $X_0$ is Cohen-Macaulay of codimension $2$ in affine space, so determinantal (Hilbert-Burch). When also $\dim X_0\le 3$ it is smoothable; see Schaps' paper in Am. J. Math., vol. $99$, for all this.</p>