Question: 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?

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).

Of course, maybe there is some easy reference on deformations of singularities which answers this? (Hopefully)

Example: I actually have a very specific example in mind whose presentation is given below:

$$\frac{\mathbb{F}_2[[a,b,c,d]]}{(bc, d^2 + ab^2, dc)}.$$

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.

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).

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).


1 Answer 1


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.

  • $\begingroup$ Thanks! I'll think about this and after I have a chance to do that, probably accept this answer. $\endgroup$ Jan 20, 2012 at 11:41

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