Existence of log canonical modifications and its applications

TL;DR

This paper proves the existence of log canonical modifications for pairs, generalizing previous results, and explores their applications in minimal model theory, including extremal rational curves.

Contribution

It establishes the existence of log canonical modifications under broad conditions and constructs dlt blow-ups with enhanced properties, advancing minimal model program techniques.

Findings

Proves existence of log canonical modifications for normal pairs.

Constructs semi-log canonical modifications for demi-normal pairs.

Analyzes lengths of extremal rational curves.

Abstract

The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some suitable assumptions. It recovers Kawakita's inversion of adjunction on log canonicity in full generality. We also discuss the existence of semi-log canonical modifications for demi-normal pairs and construct dlt blow-ups with several extra good properties. As applications, we study lengths of extremal rational curves and so on.