There are a number of techniques in algebraic geometry that can be used to show that a given reducible (often genus-zero) curve $C$ in a smooth variety $X$ becomes smooth and irreducible after a (generic) deformation. Such techniques are used, for instance, to show that (for smooth varieties in characteristic zero) rational chain-connectedness implies rational connectedness.

However, suppose you have a genus-zero reducible curve that you suspect *cannot* be deformed (under a generic deformation) to a smooth rational curve--or even an irreducible rational curve. (More precisely, you suspect that for a given irreducible component of the Kontsevich moduli of stable maps, the general point corresponds to a reducible curve.) Are there any obvious criteria that would allow you to show this?