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In the paper Families of Rationally Connected Varieties, the authors consider a map $X\rightarrow B$, where $X$ is obtained by taking disjoint copies of $B$ mapping isomorphically to $B$ and identifying pairs of points on different components of $X$ lying over the same point of $B$.

The paper says that it is easy to check that the morphism can be smoothed. What theorems need to be cited to make this work? In particular, I think I see how to get a formal family of deformations, but how do we get a family over some finite type scheme $S$?

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    $\begingroup$ First of all, there is a more precise formulation of this step in the article with de Jong. Secondly, just choose a sufficiently ample invertible sheaf on the domain curve, say $\mathcal{L}$ on $C$, with associated projective embedding $C\hookrightarrow \mathbb{P}^n$. Then let $C\to B\times \mathbb{P}^n$ be the diagonal embedding. Deformations of $C$ as a curve with a morphism to $B$ lift to deformations of $C$ as a closed subscheme of $B\times \mathbb{P}^n$. So the Hilbert scheme gives $S$. $\endgroup$
    – Jason Starr
    Commented Aug 15, 2016 at 12:03

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