Let $k$ be algebraically closed. The claim is that there exists a functor
$\{$Finitely generated extensions of $k$ of transcendence degree $1\} \rightarrow \{$Smooth, connected, proper, integral curves over $k \}$
Sending $K/k(T)/k$ to the fibre product $C=A^{-1} \times_{A^{+1/-1}} A^{+1}$. Where $A^{\ast}$ is the Spectrum of the normal closure of $k[T^{\ast}]$ in $K$.
How can I see that this gluing construction is proper/projective (all the rest should be easy)?