Liftings of ideals in positive characteristic to those in characteristic zero:Surface case

In this paper, we introduce the notion of a characteristic-zero lifting of an object in positive characteristic by means of ``skeletons’'. Using this notion, we relate invariants of singularities in positive characteristic to their counterparts in characteristic zero. As an application, we prove that the set of log discrepancies for pairs consisting of a smooth surface and a multi-ideal is discrete. We also show that the set of minimal log discrepancies and the set of log canonical thresholds of such pairs in positive characteristic are contained in the corresponding sets in characteristic zero. Another application is the construction of Campillo’s complex model of a plane curve in positive characteristic via the skeleton lifting method.