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

TL;DR

This paper introduces a method called skeleton lifting to relate singularity invariants in positive characteristic to those in characteristic zero, with applications to surface singularities and plane curves.

Contribution

It develops the notion of skeleton-based characteristic-zero liftings for objects in positive characteristic, connecting invariants across characteristics.

Findings

Discreteness of log discrepancies for smooth surface pairs with multi-ideals.

Containment of positive characteristic invariants within characteristic zero sets.

Construction of Campillo's complex plane curve model via skeleton lifting.

Abstract

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.