Let $f:X\rightarrow Y$ be a morphism of schemes with smooth curves as fibers. Let $g:X\rightarrow Z$ be a family of smooth or nodal curves with $Z$ a regular scheme. Does the push-out $Z\coprod_X Y$ exist (at least as algebraic stack)? I saw existence of the push-out as algebraic space when one has $f$ a closed immersion and $g$ a finite morphism (fibers of dimension zero). I am wondering what is going on if the fibers have dimension one.

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yourquestion. In what sense is the quotient a stack (algebraic or not), rather than a (set-valued) presheaf, sheaf, etc.? If you tell us the stack structure, MO users can try to help you determine whether or not your stack is algebraic. However, I see no natural stack structure beyond the obvious structure of a (set-valued) presheaf, or maybe a sheafification of that presheaf. $\endgroup$ – Jason Starr Jan 6 '15 at 17:56