# Lech-Mumford constant and stability of local rings

## TL;DR

This paper explores the concept of local semistability in singularities, demonstrating that semistable singularities are log canonical and introducing the Lech-Mumford constant as a key invariant in this context.

## Contribution

It advances the understanding of Mumford's local semistability, establishes new links with log canonical singularities, and develops the theory of the Lech-Mumford constant with new examples.

## Key findings

- Semistable singularities are log canonical under mild conditions.
- Introduced and studied the Lech-Mumford constant as an invariant.
- Provided new examples of semistable and unstable singularities.

## Abstract

We study further Mumford's notion of local semistability and, in particular, show that semistable singularities are log canonical under mild assumptions. We provide many new examples of semistable and unstable singularities. More generally, we develop the theory of the Lech-Mumford constant, an invariant defined as an optimal constant in the Lech inequality.