  Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by
problems in combinatorics and hypergraphs, we introduce the notion of the
symmetric geometric rank of a symmetric tensor. This quantity is equal to the
codimension of the singular locus of the hypersurface associated to the tensor.
We first derive fundamental properties of the symmetric geometric rank. Then,
we study the space of symmetric tensors of prescribed symmetric geometric rank,
which are spaces of homogeneous polynomials whose corresponding hypersurfaces
have a singular locus of bounded codimension.

Algebraic Geometry