Etaquotients

Etaquotients are a concept in the field of algebraic geometry, specifically within the study of algebraic surfaces. They were introduced by Miles Reid in the 1980s as a tool to understand the singularities of surfaces. An etaquotient is a quotient of a surface by a finite group action, where the group acts freely on the complement of a finite set of points. These points are called the fixed points of the action.

The etaquotient of a surface is a way of compactifying the quotient space, which is not necessarily compact. The compactification is achieved by adding the fixed points of the action as points at infinity. The resulting space is a surface with singularities at the fixed points.

Etaquotients have been used to study the topology of algebraic surfaces, as well as the geometry of their singularities. They have also been used to construct examples of surfaces with interesting properties, such as surfaces with rational singularities or surfaces with non-rational singularities.

In recent years, etaquotients have been generalized to higher dimensions, leading to the study of etaquotients of algebraic varieties. This has opened up new avenues of research in algebraic geometry.