Conic bundles

Consider a conic bundle over a smooth incomplete curve $C$, i.e. a smooth surface $S$ with a proper surjective morphism to $C$ such that the push-forward of the structure sheaf of $S$ coincides with the structure sheaf of $C$, and the anticanonical class of $S$ is ample over $C$. I will tell about a necessary and sufficient condition for the existence of an extension of this conic bundle to the completion of $C$. The talk is based on a joint work in progress with V. Vologodsky.