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Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure dimensional, and for all $t\in T$, $t\not =0$, the fibers $X_t:=f^{-1}(t)$ are smooth curve singularities. Is it true then that $X_0$ is reduced at $x$?

I have a counterexample for this question in the case $X$ is reduced but not pure-dimensional (for that example, $X_t$ are smooth for all $t\not =0$, but they have two isolated points, whereas $X_0$ has an embedded non-reduced point at $0$).

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    $\begingroup$ I think I am misunderstanding the question: what about e.g. elliptic surfaces with multiple fibres? $\endgroup$
    – user5117
    Commented Feb 13, 2014 at 15:38
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    $\begingroup$ Also, the phrase "smooth curve singularities" is confusing; do you mean that the fibers are smooth? $\endgroup$
    – user5117
    Commented Feb 13, 2014 at 15:42
  • $\begingroup$ @Artie: $f:X \rightarrow T$ has smooth generic fibers who all are curve singularities. The special fiber maybe not smooth, but my question is, whether the smoothness of the generic fibers ensure for the reducedness of the special fiber (together with the pure-dimensionality of $X$, i.e., the generic fibers has no isolated points)? $\endgroup$
    – Trinh
    Commented Feb 13, 2014 at 16:06
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    $\begingroup$ Dear @user46910, I suppose I am objecting to saying that a smooth fiber is a "curve singularity". That seems needlessly complicated --- why not just say it is smooth? About your question, I am asserting there are fibred surfaces where the generic fibre is smooth, but some (scheme-theoretic) fibre is nowhere reduced. Is that the kind of example you are looking for? $\endgroup$
    – user5117
    Commented Feb 13, 2014 at 16:10
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    $\begingroup$ To be more precise, e.g., any elliptic pencil on an Enriques surface. $\endgroup$
    – Alex Degtyarev
    Commented Feb 13, 2014 at 16:52

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The fact that $(X,x)$ is reduced and pure dimensional is not enough. For instance, consider $X$ as the union of the two planes $x_1=x_2=0$ and $x_3=x_4=0$ in $\mathbb C^4$ and take $f(x_1,x_2,x_3,x_4)=x_1+x_2+x_3+x_4$. Then $(X_0,0)$ is not reduced.

It is true when $(X,x)$ is Cohen-Macaulay and the fibre $(X_0,x)$ has isolated singularity. Since $(X,x)$ is Cohen-Macaulay of dimension 2, $(\mathbb C,0)$ is smooth of dimension 1 and the fibre $(X_0,x)$ has dimension 1, the map $f$ is flat. It follows that $f$ is a non zero divisor in the local ring of $(X,x)$ and hence $(X_0,x)$ is also Cohen-Macaulay. Now Serre's criterion $R_0+S_1$ gives that $(X_0,x)$ is reduced.

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