Nilpotent algebras and affinely homogeneous surfaces (Q443948)

To a nilpotent commutative algebra \(N\) of finite dimension over a field of zero characteristic one associates a smooth algebraic subvariety \(S \subset N\), whose degree is the nil-index and whose codimension is the dimension of the annihilator \(A\) of \(N\). The case when \(N\) is graded and \(A\) is 1-dimensional is studied in detail, in particular the question to what extent the hypersurface \(S\) determines the algebra \(N\). A key example with nil-index 5 is given.