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Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber is smooth and the special fiber is a reduced, simple normal crossings divisor. Suppose further that $X$ is a Cohen-Macaulay, projective variety (in particular, integral). Is this sufficient to conclude that $X$ is a regular $k$-scheme? If not, is there any known bound on the codimension of the singular locus of $X$?

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  • $\begingroup$ Do you really mean upper bound on the codimension? Usually, you'd want the dimension to be small, i.e. the codimension to be large. $\endgroup$ – R. van Dobben de Bruyn Apr 8 '18 at 17:06
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    $\begingroup$ Take $R=k[[t]]$ and $X=\mathrm{Proj}\,(R[x,y,z]/(xy-t^2z))$. Then $X$ is singular at $t=x=y=0$. $\endgroup$ – Laurent Moret-Bailly Apr 8 '18 at 20:11

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