The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms.

Suppose k is an algebraically closed field of characteristic 0,
A, B, and C are smooth varieties over k,
and f: A→B and g: A→C are closed regular embeddings of varieties.
Although the category of schemes (or varieties over k) is not cocomplete,
Corollary 3.9 in Karl Schwede's paper [Gluing schemes and a scheme without closed points](http://personal.psu.edu/kes32/Papers/SchemeWithoutPoints.pdf)
says that the pushout of f and g in the category of schemes exists.
Is this pushout a (not necessarily irreducible) variety over k,
i.e., a reduced separated scheme of finite type over k?