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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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You could look into "Deformations of rational singularities and their resolutions" by O. Riemenschneider. This deals with contraction of curves in families. There is also an article on contraction of curves by J. Lipman called "Rational singularities with applications to algebraic surfaces and unique factorization". Here the author generalizes criterion for contration of curves as given in Artin's paper "Some numerical criteria for contractability of cyrves in algebraic surfaces". May be you already know about these references. I do not know if this helps.

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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.

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