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Let $R$ be a regular local ring in mixed characteristic. Moreover, I assume that $R$ is the local ring of a point on a smooth $\mathbb Z_p$-scheme and that $R/pR$ is regular. ($\mathbb Z_p$ is the localization of $\mathbb Z$ at $(p)$).

Let me call a projective model of $R$ a pair $(X,x)$, where $X$ is projective over $\mathbb Z_p$, $R\simeq\mathcal O_{X,x}$. Clearly, under my assumptions $R$ has a projective model.

Question: is it always possible to construct a model such that the special fiber $X_p$ is "good enough"? For example, is it always possible to have $X_p$ smooth away from codimension two? How about reduced away from codimension two?

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    $\begingroup$ It is not possible to have $X_p$ smooth away from codimension two: in the relative dimension $1$ case this amounts to asking if you can have $X_p$ to be smooth and this is not always possible, e.g. for elliptic curves. $\endgroup$
    – naf
    Commented Sep 20, 2014 at 6:55

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