Variety of Jordan algebras in small dimensions

The variety $\mathcal{J}or_n$ of Jordan unitary algebra structures on $\bf{k}^n$, $\bf{k}$ an algebraically closed field with char $\bf {k}\neq 2$, is studied, as well as infinitesimal deformations of Jordan algebras. Also we establish the list of $\rm{GL}_n$-orbits on $\mathcal{J}or_n$, $n=4,5$ under the action of structural transport. The numbers $jor_4$ and $jor_5$ of irreducible components are 3 and 6 respectively; a list of generic structures is included.