Let $X\_i$ be a smooth projective variety with a smooth divisor $D\_i$ for $i=1,2$.
Suppose that $D\_1$ is isomorphic to $D\_2$. Then does it make sense to construct a normal crossing variety $X=X\_1 \cup\_D X\_2 $ by 'pasting' $D\_1$ and $D\_2$, where '$\cup\_D$' means pasting along $D\_1$ and $D\_2$?


If the answer is yes. Suppose that there is ample divisor $H\_i$ on $X\_i$ such that 
$H\_1|\_{D\_1}$ is lineary equivalent to $H\_2|\_{D\_2}$. Then is the normal crossing variety $X$ necessarily projective?



Constructing normal crossing varieties
Q1: Yes. For affine schemes, the gluing construction you need is just the [fiber product of rings](https://mathoverflow.net/a/101178/18060). Because gluing two affines in this way produces an affine, you can glue two schemes and get a scheme by taking an affine cover of each.


Q2: Yes, and the proof is more or-less completely explicit. Choose a high enough $n$ such taht $\mathcal O(n H\_i)$ is a projextive embedding of $X\_i$ and $\mathcal I\_{D\_i} \otimes \mathcal O(n H\_i)$ is globally generated with no higher cohomology, for both $i$.


Then in the embedding $X\_i \to \mathbb P^{N\_i}$, $D\_i$ is contained in some linear subspace cut out by the functions in $H^0(X\_i, \mathcal I\_{D\_i} \otimes \mathcal O(n H\_i))$. Because they generate, in fact $D\_i$ is the intersect of $X\_i$ with that linear subspace. Because the map $H^0(X\_i, \mathcal O(nH\_i)) \to H^0(X\_i, \mathcal O\_{D\_i} (nH\_i))$ is surjective, the emedding of $D\_i$ into that linear subspace is just the embedding coming from the very ample line bundle $nH\_i$. This embedding is the same whether $i=1$ or $2$.


So $X\_1$ and $X\_2$ are expressed as two projective varieties with an isomorphism between their restrictions to two linear subspaces. We can easily embed the two projective spaces in a higher-dimensional projective space such that their intersection is exactly that linear subspace. Then the union of $X\_1$ and $X\_2$ in that higher space will clearly be $X\_1 \cup\_D X\_2$, and hence it is a projective variety.


