Extracting non-canonical places

TL;DR

This paper constructs specific birational models for log canonical pairs with divisorial valuations in a finite set, and explores their relationships and applications to deformations of singularities.

Contribution

It proves the existence of projective birational models with exceptional divisors corresponding to given valuations and analyzes their interrelations and deformation implications.

Findings

Existence of models with prescribed divisorial valuations.

Relations between different such models.

Application to deformation theory of singularities.

Abstract

Let $(X,B)$ be a log canonical pair and $\mathcal{V}$ be a finite set of divisorial valuations with log discrepancy in $[0,1)$. We prove that there exists a projective birational morphism $\pi \colon Y\rightarrow X$ so that the exceptional divisors are $\mathbb{Q}$-Cartier and correspond to elements of $\mathcal{V}$. We study how two such models are related. Moreover, we provide an application to the study of deformations of log canonical singularities.