Forms of low slice rank

The slice rank of a polynomial f of degree d is the minimum natural number r such that f can be written as a sum of r products of a polynomial of degree d-1 by a linear form. An important property of slice rank is its lower semi-continuity.

Although the value of the general slice rank is classically known, no much is known about the equations of its secant varieties.  We provide some geometric and set-theoretic properties of varieties parametrizing forms with low slice rank, focusing on cubic forms.