This question is in some sense a follow up to a related question [Is a normal proper relative curve over a DVR projective?](https://mathoverflow.net/questions/202109/is-a-normal-proper-relative-curve-over-a-dvr-projective)


Let $R$ be a Dedekind domain, let $S := \mathrm{Spec}(R)$, and let $X \rightarrow S$ be a proper flat morphism with $X$ a regular algebraic space whose $S$-fibers are purely of dimension $1$ (and hence schemes, e.g. by <http://stacks.math.columbia.edu/tag/0ADD>). Is $X \rightarrow S$ actually projective, so that $X$ is a scheme itself? 


The projectivity in the case of a scheme $X$ is an old result of Lichtenbaum.



Is every proper regular relative algebraic space curve over a Dedekind domain projective?
Yes. The task is to show that $X$ is a scheme (as then Lichtenbaum's result may be applied). By standard "spreading out" arguments, we may assume $S = {\rm{Spec}}(R)$ for a discrete valuation ring $R$, say with fraction field $K$, residue field $k$, and maximal ideal $\mathfrak{m}$. The special fiber $X\_k$ is a scheme. Let $\{C\_1, \dots, C\_n\}$ be the irreducible components of $X\_k$ (say with reduced structure). 


Let $x\_i \in C\_i$ be a closed point, and let $U\_i \rightarrow X$ be a residually trivial etale affine neighborhood of $x\_i$ that is a scheme, so $U\_i$ is a regular affine $R$-curve (i.e., flat finite type over $R$ with fibers of pure dimension 1). We may shrink $U\_i$ so that its special fiber misses each $k$-finite $C\_i \cap C\_j$ for $i \ne j$, which is to say that $(U\_i)\_k$ lands in $C\_i$. By going-down for flat morphisms, applied to $R \rightarrow O\_{U\_i,x\_i}$, we can choose a closed point $u\_i \in (U\_i)\_K$ whose closure in $U\_i$ contains $x\_i$. Thus, the image $\xi\_i \in X\_K$ is a closed point whose closure in $X$ contains $x\_i$.


Since $X$ is regular of pure relative dimension 1 over $R$, the closure $D\_i$ of $\xi\_i$ in $X$ has *invertible* ideal sheaf $\mathscr{I}\_{D\_i}$. (Indeed, such invertibility may be checked on an etale scheme cover of $X$, where it becomes the fact that a height-1 prime in a regular domain in invertible, as regular local rings are UFD's.) 


Let $\mathscr{L}$ denote the tensor product of the inverse sheaves $\mathscr{I}\_{D\_i}^{-1}$. Since each $D\_i$ is $R$-flat, the formation of $\mathscr{L}$ commutes with base change on $R$, such as passage to the special fiber. By the theory of algebraic curves, $\mathscr{L}\_k$ is thereby seen to be ample on $X\_k$. Hence, for any coherent sheaf $\mathscr{F}$ on $X$, for all sufficiently large $m$ we have that $\mathscr{F} \otimes \mathscr{L}^m$ has vanishing degree-1 cohomology on $X\_k$. 


If $\mathscr{F}$ is $R$-flat and $\pi \in R$ is a uniformizer then we have a short exact sequence
$$0 \rightarrow \mathscr{F} \otimes \mathscr{L}^m \stackrel{\pi}{\rightarrow}
\mathscr{F} \otimes \mathscr{L}^m \rightarrow \mathscr{F}\_k \otimes \mathscr{L}\_k^m \rightarrow 0$$
whose associated cohomology sequence gives the vanishing of ${\rm{H}}^1(X, \mathscr{F} \otimes \mathscr{L}^m)=0$ for such large $m$ (due to Nakayama and $R$-finiteness of this H$^1$). Applying this with $\mathscr{F}$ equal to the ideal sheaf on $X$ of a (varying) closed point of $X\_k$, we see that for some large $m\_0$ the line bundle $\mathscr{L}^{m\_0}$ is generated by global sections. By replacing $m\_0$ with a big multiple, the formation of the $R$-finite (free) module of global sections commutes with reduction modulo $\mathfrak{m}$.


Now we get a natural $R$-map $f:X \rightarrow {\rm{Proj}}(\Gamma(R, \mathscr{L}^{m\_0}))$ whose formation commutes with reduction modulo $\mathfrak{m}$, 
and on the special fiber it is quasi-finite since $\mathscr{L}\_k$ is ample on $X\_k$ with $m\_0$th-power generated by global sections. For any map of finite type between noetherian algebraic spaces, the quasi-finite locus is open on the source (as may be checked etale-locally, where it reduces to the known analogue for schemes), so the open quasi-finite locus $U \subset X$ of $f$ contains $X\_k$. But $X$ is $R$-proper, so this forces $U=X$. Hence, $f$ exhibits $X$ as separated and quasi-finite over a scheme, so $X$ is a scheme (by a theorem of Knutson).


[The preceding is not optimal, since the conclusion that $X$ is a scheme, even quasi-projective, should be true with "proper" relaxed to "separated, flat, and finite type".


