# The variety of $2$-dimensional algebras over an algebraically closed field

## TL;DR

This paper classifies 2-dimensional algebras over an algebraically closed field, analyzes their degenerations, and describes the structure of subvarieties including flexible and bicommutative algebras, identifying rigid algebras and irreducible components.

## Contribution

It provides a comprehensive classification and structural analysis of 2-dimensional algebras and their subvarieties, including degenerations and rigid algebras.

## Key findings

- Classification of 2-dimensional algebras up to isomorphism.
- Description of degenerations and closures of algebra series.
- Identification of rigid algebras and irreducible components.

## Abstract

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the variety under consideration. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.