The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras

TL;DR

This paper provides a detailed geometric classification of complex 5-dimensional nilpotent commutative -algebras, revealing their variety structure and irreducible components.

Contribution

It introduces a comprehensive geometric framework for classifying these algebras, identifying their variety dimension and decomposition into irreducible components.

Findings

The variety of such algebras has dimension 24.

It decomposes into 10 irreducible components.

Identifies specific families and rigid algebras within the classification.

Abstract

We give a geometric classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension $24$ and decomposes into $10$ irreducible components determined by the Zariski closures of a two-parameter family of algebras, three one-parameter families of algebras, and $6$ rigid algebras.