Let $S$ be a Dedekind scheme with function field $K=K(S)$ and $C$ a projective regular curve over $K$, so we can fix certain closed embedding $e:C \subset \mathbb{P}^n_K$.
Let compose this embedding with canonical map $f: \mathbb{P}^n_K \to \mathbb{P}^n_S$ and consider the Zariski closure $Z_C:=\overline{f\circ e(C)}$ in $\mathbb{P}^n_S$ of the image of $C$.
Q ( the "initial" one): Is it true that $Z$ has dimension $2$ and if yes, how to see it?
(#Edit: Think so, if the sketched argument in "Idea" below works)
Can the statement be generalized (#1) to higher dimensions in following way: Say $X \subset \mathbb{P}^n_K$ is an irreducible subscheme of demension $d$ and we take as before the closure $Z_X:=\overline{f\circ e(X)}$ of the image of $X$ in $\mathbb{P}^n_S$. What can we say about it's dimension?
So the point becomes if that's a specific curve-to-surface feature, or does it hold in higher dimensions as well?
an "idea"/ plagiarism: If I'm not missing something the same argument as Daniel Loughran gave here should go through in this problem as well, should't it?
Namely, the induced projection map $p:Z_X \to S$ is dominant, so flat, since we are over Dedekind domain $S$ and by assumption the generic fiber has dimension $d$ of $X$, so by this flatness argument $ Z_X$ should have dimension $d+1$ if we add up fiber and base dimensions (which works for dominant flat maps since flatness assures that the fiber dimension stays constant). Is the exposed argumentation correct so far?
If the sketched argument in the Idea above works, then this leads me to the "natural" attempt to generalize this this question once more by now dropping the flatness feature, which as $S$ Dedekind, was in above formulations "donated for free" to us :
Generalisation #2: What can we say about the dimension of the closure $Z_X \subset \mathbb{P}^n_S$ if $S$ would be instead an arbitrary irreducible locally Noetherian scheme with field of fractions $K$? (Note that induced map $Z_X \to S$ would be still dominant, so surjective due to properness.
Best Answer
I'll interpret your terminology "Dedekind scheme" to mean "regular integral locally Noetherian scheme of dimension one" (or dimension $\leq 1$ if you replace $d + 1$ with $d + \operatorname{dim} S$ in your notation).
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When $S$ is a Dedekind scheme, the answer to your question (for $Z$ of arbitrary dimension) is yes via flatness as you argue (e.g.Tag 0D4J and Tag 0AFE). (Small comment: for flatness of $Z_X \rightarrow S$, being dominant is not enough, e.g. there could be an embedded point of $Z_X$ lying over a closed point of $S$ like $\operatorname{Spec} k[x,y]/(xy, y^2) \rightarrow \operatorname{Spec} k[x]$ for a field $k$. But it will be ok in your situation, where you are taking scheme-theoretic closure.)
Remark: The properness is being used here (but projectivity is not important). Suppose $S = \operatorname{Spec} \mathbb{Z}_{(p)}$ and replace $\mathbb{P}^n_S$ in your notation with $\mathbb{A}^2_S = \operatorname{Spec} \mathbb{Z}_{(p)}[x,y]$. Consider the curve $C = \operatorname{Spec} \mathbb{Q}[x,y]/(p x - 1)$ in the generic fiber. The closure of $C$ in $\mathbb{A}^2_S$ is $\operatorname{Spec} \mathbb{Z}_{(p)}[x,y]/(px - 1)$. But this scheme has empty fiber over the closed point of $S$.
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Next for your Generalisation #2. Probably we should at least assume $S$ is finite dimensional (as $S$ could have infinite dimension like in Tag 02JC). I will also assume $S$ is reduced. (This is harmless because the reduced closed subscheme $S_{\mathrm{red}} \rightarrow S$ is a universal homeomorphism, and since scheme-theoretic closure along quasi-compact morphisms has underlying set coinciding with set-theoretic closure).
I will add some some more hypotheses: assume $S$ is universally catenary, Jacobson, and admits a (necessarily unique) "dimension function" $\delta \colon S \rightarrow \mathbb{Z}$ which sends closed points to $0$ (as in Tag 02QK). This dimension function $\delta$ sends any point to the (topological) dimension of its closure.
These additional hypotheses on $S$ are satisfied, for example, if (a) $S$ is locally finite type over a field or if (b) $S$ is locally of finite type over a Dedekind domain with infinitely many primes.
Then Tag 02QO implies that what you ask for indeed holds. More precisely/generally: Suppose $X$ is a scheme which is locally of finite type over $S$. Suppose $Z \subseteq X_K$ is an irreducible closed subscheme of the generic fiber, with closure $\overline{Z} \subseteq X$. Then $\operatorname{dim} \overline{Z} = \operatorname{dim} Z + \operatorname{dim} S$.
Remark: Note that there are no properness hypotheses in the result above. This result is false in general if $S$ is not Jacobson (e.g. $S = \operatorname{Spec} \mathbb{Z}_{(p)}$ as in the remark above).
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I don't have an answer for your Generalisation #2 outside the cases covered above.