A categorical quotient in the category of dense constructible subsets

TL;DR

This paper demonstrates that a previously known example of a torus action without a quotient in algebraic varieties does have a quotient when considering dense constructible subsets, expanding the understanding of quotients in algebraic geometry.

Contribution

It introduces the concept of categorical quotients within the category of dense constructible subsets, providing a new perspective on quotient existence beyond algebraic varieties.

Findings

The example admits a quotient in the dense constructible subsets category.

Answers a question posed by A. Bialynicki-Birula.

Expands the framework for understanding quotients in algebraic geometry.

Abstract

A. A'Campo-Neuen and J. Hausen gave an example of an algebraic torus action on an open subset of the affine four space that admits no quotient in the category of algebraic varieties. We show that this example admits a quotient in the category of dense constructible subsets and thereby answer a question of A. Bialynicki-Birula.