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From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings.

Now, if I start in the projective world I would like to be able to stay there under pushout. However, it seems that the answer to this questions is a counterexample in which one glues two copies of $\mathbb{P}^3$ along two different embedded curves. So naturally, I have a question if there is a condition under which the pushout of embeddings of projective varieties is again projective.

The main example that I would like to understand is as follows. Let $i:D\hookrightarrow X$ be a particular embedding of a normal crossing divisor into a smooth projective variety $X$ over $\mathbb{C}$. Is $X\cup_{i,D,i} X$ projective again?

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    $\begingroup$ Section 5 here, especially Example 36. It shows that you need to be careful with how you glue the two copies of $D$. $\endgroup$
    – Misha
    Commented Jan 5, 2020 at 13:23
  • $\begingroup$ It seems that in the example 36, they reparametrize the embeddings of the divisor. Is there any indication why a problem might occure if I take equal embeddings? $\endgroup$
    – Arkadij
    Commented Jan 5, 2020 at 13:40
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    $\begingroup$ See also D. Ferrand, section 6.3 in smf.emath.fr/publications/conducteur-descente-et-pincement $\endgroup$
    – Laurent Moret-Bailly
    Commented Jan 5, 2020 at 13:49
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    $\begingroup$ @ArkadijBojko: In this case, there is no problem and the amalgam will be projective. You can use Ferrand's paper as a reference. I suggest you revise your question clarifying the definition of amalgams you are interested in. $\endgroup$
    – Misha
    Commented Jan 5, 2020 at 14:50

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