Let $X$ be a smooth projective variety. Is there a criterion (apart from the definition) for the existence of a projective curve $C$ and a proper surjective morphism $\pi:X \to C$?
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Another interesting theorem in this direction is Castelnuovode Franchis theorem. It says that if you have two linearly independent holomorphic 1forms $\omega_1,\omega_2$ with $\omega_1\wedge\omega_2=0$ on $X$, then there exists a morphism $f:X\to C$ with $C$ a smooth curve of genus at least 2 and forms $\omega_i'$ on $C$ such that $\omega_i=f^*\omega_i'$. 


Yes. This is equivalent to the existence of a nontrivial divisor $D$ on $X$ such that there exists two distinct members of the linear system $D\ni D_1,D_2$ that are disjoint: $D_1\cap D_2=\emptyset$. 


By a theorem of Gromov and Schoen if the fundamental group of X is a proper amalgamated product or HNN extension then X maps surjectively to a curve . 

