Let $X$ be a $R$-scheme, where $R$ is a dvr. Suppose that the reduction of $X$ (over the closed point of $\mathrm{Spec} \ R$) is smooth and projective. Does this imply that $X$ is smooth and projective? Smoothness might get messed up by the generic fibre, but what about projectiveness?
Is $X\to \mathrm{Spec} R$ projective?