Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra \({\mathcal{N}}\) of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety \({S\subset\mathcal{N}}\) can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator \({\mathcal{A}}\) of \({\mathcal{N}}\). In case \({\mathcal{N}}\) admits a grading, the surface S is affinely homogeneous. More can be said if \({\mathcal{A}}\) has dimension 1, that is, if \({\mathcal{N}}\) is the maximal ideal of a Gorenstein algebra. In this case two such algebras \({\mathcal{N}}\), \({\tilde{\mathcal{N}}}\) are isomorphic if and only if the associated hypersurfaces S, \({\tilde S}\) are affinely equivalent. If one of S, \({\tilde S}\) even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of \({\mathcal{N}}\) does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.