I am interested in constructing a morphism out of a blown down variety.
Let $V$ be a scheme, $U\hookrightarrow V$ an open immersion. Let $\widetilde V$ be a blow-up of $V$, $\widetilde U$ its restriction to $U$.
Question: when does $U\leftarrow\widetilde U\rightarrow \widetilde V$ admit a pushout in the category of schemes? One would also hope that the pushout is a proper modification of $V$ with $U$ an open subscheme.
Note: we certainly can't expect $V$ to be the pushout, even if the blow-up is surjective with centre the closure of a subscheme of $U$ (example: small resolution of 3-fold ODP). But the toric picture suggests that one can algorithmically construct some pushout, at least in the category of toric varieties.