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 says that the pushout of f and g in the category of schemes exists. Is this pushout a variety over k?