  Let $k$ be an algebraically closed field. Fix integers $n$ and $b$ with
$n\geq 3$ and $1\leq b\leq n-1.$ Let $T^d_k$ be the moduli space of
hypersurfaces $[F]$ in $\mathbb{P}^n_k$ of degree $l$ whose singular locus
contains a subscheme of dimension $b$ with Hilbert polynomial among the Hilbert
polynomials of $b$-dimensional integral closed subschemes of $\mathbb{P}^n$ of
degree $d$. We prove that when $l$ is sufficiently large and $2\leq d\leq
\frac{l+1}{2},$ any irreducible component $Z$ of $T^d_k$ satisfies $Z=T^1_k$ or
$\dim Z<\dim T^1_k.$

Algebraic Geometry