Let $X$ be a proper geometrically integral $\mathbb{F}\_p$-scheme.


Assume that $X$ is the special fiber of a proper flat $\mathbb{Z}\_p$-scheme with a smooth generic fiber and that for each point $x\in X$ we have $\dim\_{\kappa (x)}(\Omega \_{X/\mathbb{F}\_p, x}\otimes\_{\mathcal{O}\_{X, x}} \kappa (x))\leq 1+\dim(X)$.


Is $X$ the special fiber of a regular proper flat $\mathbb{Z}\_p$-scheme?



Smoothable $\mathbb{F}_p$-variety embeds in a regular scheme
**Edit**. The original post is likely wrong: it would be better to work with a $\mathfrak{g}^r\_d$ that gives a closed immersion of $C$, but which is a singular point of $\mathcal{G}^r\_d(C)$. The following example is easier conceptually to understand.


Let $k$ be a field, i.e., $\mathbb{F}\_p$. Let $C$ be a $k$-curve of genus $g>2$. Let $J$ denote $\text{Pic}^{g-1}\_{C/k}$, the Picard scheme parameterizing ideal sheaves on $C$ of degree $g-1$. Let $\Theta$ denote the Cartier divisor in $\text{Pic}^{g-1}\_{C/k}$ parameterizing ideal sheaves $\mathcal{L}$ such that $H^1(C,\mathcal{L})$ is nonzero (by Serre duality, this is the same as nonzero $H^0$, but $H^1$ behaves better with respect to base change). This is an irreducible Cartier divisor that gives a principal polarization of $\text{Pic}^{g-1}\_{C/k}$. By the Riemann Singularity Theorem, the singular locus of $\Theta$ has codimension $c$ at least $2$ in $\Theta$ (typically the codimension equals $3=4-1$, but hyperelliptic curves have singular locus of codimension $2$). The multiple $3\Theta$ is a very ample Cartier divisor. Consider subvarieties of dimension $c-1$ that are complete intersections of $\Theta$ and $g-1$ divisors in the linear system of $3\Theta$. A typical complete intersection in this family is smooth. Let $Z$ denote a member of the family that intersects the singular locus of $\Theta$.


Assume that there exists an invertible sheaf $\mathcal{A}$ of degree $g-2$ such that the induced Abel map has image that is not contained in $\Theta$ and that is disjoint from $Z$,
$$\alpha\_{\mathcal{A}}:C\to \text{Pic}^{g-1}\_{C/k}, \ \ p \mapsto \mathcal{A}(\underline{p}).$$


Denote the blowing up of $\text{Pic}^{g-1}\_{C/k}$ along the image of $\alpha\_{\mathcal{A}}$ by,
$$ \nu:\widetilde{Pic}^{g-1}\_{C/k} \to {Pic}^{g-1}\_{C/k}.$$
Denote the strict transform of $\Theta$ by $\widetilde{\Theta}$. Denote the strict transform of $Z$ by $\widetilde{Z}$.


Finally, let $X$ be the blowing up of $\widetilde{Pic}^{g-1}\_{C/k}$ along the closed subscheme $\widetilde{Z}$. By varying the $Z$ to a complete intersection curve that is disjoint from the singular locus of $\Theta$, there are generizations / deformations of $X$ that are smooth. Moreover, on open affines of $X$, there are embeddings as hypersurfaces in a regular scheme. However, since the theta divisor is unique up to translation (and all translations are still singular), there is no **global** embedding of $X$ as the closed fiber of a regular, flat model over a DVR.


**Original post (likely mistaken)**. I only have a few minutes, so I will try to fill in the details later. Let $C$ be a geometrically integral, at-worst-nodal curve of arithmetic genus $4$ embedded in $\mathbb{P}^3$ by the complete linear system of its dualizing (invertible) sheaf (in particular, $C$ is not hyperelliptic). Further assume that the unique quadric surface that contains $C$ is a cone over a smooth plane conic $P$. Thus, $C$ has a unique $\mathfrak{g}^1\_3$, which has multiplicity $2$ in the scheme $\mathcal{G}^1\_3(C)$. Finally, assume that $C$ has an ordinary double point at the vertex $p$ of the cone.


Now consider the blowing up of the cone at its vertex. The pullback of the Cartier divisor $C$ contains the exceptional divisor $E$ as an irreducible component of multiplicity $2$. Consider the curve $C'=\widetilde{C}+E$ such that the pullback equals $C'+E$, i.e., "subtract" one copy of $E$ from the pullback of $C$. The curve $C'$ is a prestable curve of arithmetic genus $4$. The morphism from $C'$ to $\mathbb{P}^3$ is an embedding on the strict transform $\widetilde{C}$. The projection from the vertex to the plane conic $P$ is a morphism from $C'$ to $P$ that restricts to an isomorphism on $E$. Together, these two morphisms define a closed immersion of $C'$ in $\mathbb{P}^3\times P$.


Finally, let $X$ be the blowing up of $\mathbb{P}^3\times P$ along the ideal sheaf of this closed immersion. This satisfies the local hypotheses. However, because the scheme $G^1\_3(C)$ has multiplicity $2$, it turns out that there is no global regular scheme that contains $X$ as a Cartier divisor with trivial normal bundle (as would be the case for a regular scheme with a proper, flat morphism to a DVR whose special fiber equals $X$). On the other hand, it is straightforward to deform $C$ inside the singular quadric surface, and this gives a lift whose generic fiber is smooth (the blowing up of $\mathbb{P}^3\times P$ along the ideal sheaf of a closed immersion of a smooth, projective curve of genus $4$).


There are many details missing in the above, but the key point is that, even though $C$, and thus $C'$, deform "transversally" giving a regular scheme over $\mathbb{Z}\_p$, the **embedded deformations** of $C'$ in $\mathbb{P}^3\times P$ are not formally smooth. Via blowing up, this gives rise to counterexamples.


