Varieties of nilpotent Jordan superalgebras of dimension five

TL;DR

This paper classifies all 5-dimensional complex nilpotent Jordan superalgebras, detailing their isomorphism classes, geometric structure, and degenerations using algebraic and geometric methods.

Contribution

It provides a complete classification of these superalgebras, including isomorphism representatives, geometric components, and degeneration relations.

Findings

Identified all isomorphism classes of 5D nilpotent Jordan superalgebras.

Determined the irreducible components of the algebraic varieties.

Described all degenerations and invariants between superalgebras.

Abstract

The paper is devoted to the description of the varieties of complex 5-dimensional nilpotent Jordan superalgebras. We find all representatives for the isomorphism classes, using the Jordan normal form, results of simultaneous matrix triangularization, the Jordan-Kronecker theorem for a pair of skew-symmetric bilinear forms and similar arguments developed for $\Delta$-modules by Burde and Grunewald. We also provide a complete geometric classification, determining the irreducible components of the corresponding varieties and describing all possible degenerations and non-degenerations between these superalgebras, in particular, applying some $\mathbb{Z}_2$-graded subspaces as invariants.