Irreducible Components of the Varieties of Jordan Superalgebras of Types $(1,3)$ and $(3,1)$

TL;DR

This paper classifies the irreducible components of the varieties of 4-dimensional Jordan superalgebras with specific even parts, revealing their structure and providing counterexamples to an analogue of the Vergne conjecture.

Contribution

It describes the irreducible components of these varieties and identifies a solvable rigid superalgebra challenging existing conjectures.

Findings

Variety is union of closures of orbits of 11 and 21 rigid superalgebras

Irreducible components explicitly described

Existence of a solvable rigid superalgebra contradicts the analogue of the Vergne conjecture

Abstract

We describe the variety of Jordan superalgebras of dimension $4$ whose even part is a Jordan algebra of dimension $1$ or $3$. We prove that the variety is the union of Zariski closures of the orbits of $11$ and $21$ rigid superalgebras, respectively. In both cases, the irreducible components of the varieties are described. Furthermore, we exhibit a four-dimensional solvable rigid Jordan superalgebra, showing that an analogue to the Vergne conjecture for Jordan superalgebras does not hold.