  We establish basic facts about the varieties of homogeneous polynomials
divisible by powers of linear forms, and explain consequences for geometric
complexity theory. This includes quadratic set-theoretic equations, a
description of the ideal in terms of the kernel of a linear map that
generalizes the Foulkes-Howe map, and an explicit description of the coordinate
ring of the normalization. We also prove asymptotic injectivity of the
Foulkes-Howe map.
