Stein factorization

A proper morphism $f:X\to Y$ factors as $X\stackrel{g}{\to }Z\stackrel{h}{\to }Y$, with $h$ finite and $g_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Z$; its fibers are connected. When $X$ is normal, $Z$ is normal. For a morphism from a smooth surface onto a curve, $Z$ is therefore a smooth projective curve over an algebraically closed field.