TITLE: Is a normal proper relative curve over a DVR projective?
QUESTION [5 upvotes]: Let $X$ be a connected normal scheme equipped with a proper flat morphism $f\colon X \rightarrow \mathrm{Spec }(R)$ with $R$ a discrete valuation ring and such that the fibers of $f$ are curves (i.e., purely of dimension $1$). Is $f$ necessarily projective?
The answer is 'yes' if the normality assumption is strengthened to regularity of $X$: this was proved in Lichtenbaum's thesis.

REPLY [4 votes]: This is perhaps not the answer the poster is looking for, but Proposition 3.3 of Families of rationally simply connected varieties over surfaces and torsors for semisimple groups, by de Jong--He--Starr, proves something related. Indeed, an immediate consequence of this result is that if $\pi \colon C \to S$ is a proper, flat, and finitely presented morphism of algebraic spaces of relative dimension 1, then there exists an etale cover $S' \to S$ such that the pullback $\pi' \colon C\times_S S' \to S'$ is projective. 
In particular, in the notation of the posed question: if $R$ is strictly henselian (e.g., complete with algebraically closed residue field), then $f$ is projective. 
Note that it is not even necessary to assume that $X$ is normal for this this result to hold.