# Stability, shards, and preprojective algebras

## TL;DR

This paper explores the geometric interpretation of semistability in finite-dimensional algebras and connects Nathan Reading's shards of hyperplane arrangements to the semistability structure of finite-type preprojective algebras.

## Contribution

It introduces a geometric perspective on semistability and links it to hyperplane arrangement shards, offering new insights into the structure of preprojective algebras.

## Key findings

- Semistability provides a geometric view of extension-closed subcategories.
- Shards of hyperplane arrangements relate to semistability in preprojective algebras.
- The paper clarifies the role of shards in understanding algebraic stability conditions.

## Abstract

The goal of this note two-fold. First, I draw attention to the way that semistability (in the sense of King) gives us a geometrical picture of (some of) the extension-closed abelian subcategories of a finite-dimensional algebra. Second, I describe Nathan Reading's shards of a hyperplane arrangement, and explain their relevance to understanding the semistability picture for finite-type preprojective algebras.