Finite determination conjecture for Mather-Jacobian minimal log discrepancies and its applications

TL;DR

This paper introduces the Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in arbitrary characteristic, linking jet schemes to singularity properties and providing evidence for the conjecture.

Contribution

It proposes a new conjecture relating jet schemes to singularity invariants and demonstrates its implications and evidence in various cases, advancing understanding of singularities in positive characteristic.

Findings

Conjecture is equivalent to bounded blow-ups for computing discrepancies.

Establishes basic properties of singularities assuming the conjecture.

Provides evidence for the conjecture in specific cases like hypersurfaces and 2D singularities.

Abstract

In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the boundedness of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy. We also show that this conjecture yields some basic properties of singularities; e.g. openness of Mather-Jacobian (log) canonical singularities, stability of these singularities under small deformations and lower semi-continuity of Mather-Jacobian minimal log discrepancies, which are already known in characteristic 0 and open for positive characteristic case.We show some evidences of the conjecture: for example, for non-degenerate hypersurfaces of any dimension in arbitrary characteristic and 2-dimensional singularities in characteristic not 2. We aslo give a bound of the number of the blow-ups to obtain a prime divisor which computes the Mather-Jacobian minimal log discrepancy.