Nilpotent algebras and affinely homogeneous surfaces

Abstract

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.