$\textbf{Question 1}$ (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.



Projectivity in flat families
For question 2: Let $Y\to S$ be a proper morphism with $S$ integral and noetherian (for simplicity) and $Y$ irreducible with non-empty projective generic fiber $Y\_K$ (where $K$ is the field of rational functions on $S$).
Then there exists a dense open subset $U$ of $S$ such that $Y\_U\to U$ is projective: 


Let $Y\_K\to \mathbb P^N\_K$ be a closed immersion. Let $Z$ be the scheme-theoretical closure of the image $Y\_K$ in $\mathbb P^N\_K$. Then we have a birational map $f$ from $Y$ to $Z$. The domain of definition $\Omega$ of $f$ is open and contains the generic fiber $Y\_K$. So $\Omega$ contains $Y\_V$ for some dense open subset $V$ of $S$ and we have a birational morphism $Y\_V\to Z\_V$. 


Applying the same reasonning to this morphism or to the inverse birational map, we see that there exists a dense open subset $U$ of $V$ such that $Y\_U\to Z\_U$ is an isomorphism. So $Y\_U\to U$ is projective. 


Now for all $s\in U$, $Y\_s$ is projective over the residue field $k(s)$ of $S$ at $s$. 


