Geometric classification of nilpotent Jordan algebras of dimension five

The variety JorN5 of five-dimensional nilpotent Jordan algebras structures over an algebraically closed field is investigated. We show that JorN5 is the union of five irreducible components, four of them correspond to the Zariski closure of the GL5-orbits of four rigid algebras and the other one is the Zariski closure of an union of orbits of an infinite family of algebras, none of them being rigid.