$\textbf{Question 1}$ (Generically projective over $\bar{k}$) Let $X, Y$ be schemes of finite type over a fixed field $k = \bar{k}$, and we work with $k$-schemes. Suppose $\phi: Y \rightarrow X$ is flat of finite type, with $X$ integral and $Y$ proper, irreducible. If the fibers $Y_z$, of $\phi$ over closed points $z$ in a nonempty open subset of $X$ are projective (over $k$), then does it follow that all fibers over closed points are in fact projective (over $k$).

$\textbf{Question 2}$ (Spreading out) If $\phi: Y \rightarrow \textrm{Spec } \mathbb{Z}$ is flat of finite type with $Y$ irreducible and $X$ integral, and the generic fiber $Y_\mathbb{Q} \rightarrow \mathbb{Q}$ a projective morphism. Are the fibers $Y_{\textbf{F}_p} \rightarrow \textrm{Spec } \mathbb{F}_p$ also projective morphisms? Do we have a version of "spreading out", that is, if the morphism is projective at the generic fiber then is is so over a nonempty open set.

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