Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.

Let $C_R$ be a flat regular proper model for $C$ over $R$ and take $\mathcal{X}$ the scheme-theoretic closure of $X$ in $\mathbf{P}^n_{C_R}$.

Is $\mathcal{X}$ necessarily flat over $C_R$? What if $C_R = \mathbf{P}^1_R$?

I'd expect "no" because the scheme-theoretic closure is flat when $C_K$ is replaced by the open complement of a point in a regular integral scheme of dimension $1$, whereas $C_R$ is of dimension $2$.

notthe same dimensional) then it won't be flat. Or if $\mathcal{X}$ is even finite over $C_R$ but $\mathcal{X}$ is not S2, it's not flat. $\endgroup$