Algebras with irreducible module varieties III: Birkhoff varieties

TL;DR

This paper investigates a family of affine varieties linked to classifying embeddings of finite abelian p-groups, establishing their irreducibility and the existence of dense orbits, thus advancing understanding of their geometric structure.

Contribution

It proves that these Birkhoff-related varieties are irreducible and possess dense orbits, providing new insights into their geometric and algebraic properties.

Findings

All varieties are irreducible.

Existence of dense orbits in these varieties.

Advancement in classifying embeddings of finite abelian p-groups.

Abstract

We study a family of affine varieties arising from a version of an old problem due to Birkhoff asking for the classification of embeddings of finite abelian p-groups. We show that all of these varieties are irreducible and have a dense orbit.