# Contraction of curves on surfaces

Assume we have a surface $S$ (smooth if you want), and a map $f: S \to V$ that contracts a curve $C \subset S$. What condition would give a factoring of $f$ through a contraction $c: S \to V'$ contracting $C$ but otherwise being an isomorphism?

What about the special case where $S$ is a flat family of curves and $C$ is the central fiber (all other fibers are assumed smooth). What criterion would give a factoring of $f$ through the contraction?

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The question is not precise enough: is $f$ proper? Is $C$ smooth, or at least irreducible? What is a contraction?
Any how, the answer to the last question is negative, at least in the following formulation: $g:S\rightarrow B$ is flat and proper, with connected fibers, and $C$ is one of the fibers. Then by Mumford's rigidity lemma (GIT, chap. 6, Prop. 6.1), any map $f:S\rightarrow V$ which contracts $C$ must contract every fiber - in other words, $C$ cannot be contracted by a birational morphism.