  We explore the connection between the rank of a polynomial and the
singularities of its vanishing locus. We first describe the singularity of
generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic
surfaces with isolated singularities are known to fall into 22 singularity
types. We compute the rank of a cubic surface of each singularity type. This
enables us to find the possible singular loci of a cubic surface of fixed rank.
Finally, we study connections to the Hessian discriminant. We show that a cubic
surface with singularities that are not ordinary double points lies on the
Hessian discriminant, and that the Hessian discriminant is the closure of the
rank six cubic surfaces.
