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Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$ is a regular sequence in $S$.

Example: $A=k[x]/(x^2)$, $S=k[[x,y]]/(xy)$, $f_1=(x-y)$.

This definition is closely related to whether $A$ is smoothable, i.e whether you can flat deform $A$ to $k^n$, or equivalently, deform $Spec(A)$ to a smooth scheme of $n$ distinct points. The smoothing problem is of course well-studied, for example one can look at this new paper by Erman and Velasco for some new results on obstructions to smooothing and references on works by Artin, Hartshorne, Iarrobino, etc.

My intuition is that having reduced embedded deformation is weaker than smoothability. I am interested in obstruction to reduced embedded deformation, but could not find any in the literature (I looked briefly at Iarrobino-Emsalem paper and Vistoli's note, but I might have missed something obvious). It feels like this would be known by the experts, so this question sounds exactly like the kind one should ask on MO:

Question: How close are having reduced embedded deformation and smoothability? What are the known obstruction criteria for the former (preferably with concrete examples)?

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  • $\begingroup$ @VA: You are right, my reason is the example also illustrates the smoothing situation: let t=x-y, then S is flat over k[[t]], S/(t)=A, and $S/(t-\alpha) \cong k^2$ for all $\alpha \in k$. $\endgroup$ – Hailong Dao Apr 17 '10 at 2:59

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