On the moduli space of hypersurfaces singular along a subscheme of large dimension but small degree

Let $k$ be an algebraically closed field. Fix integers $n$ and $b$ with $n\ge 3$ and $1\le b\le n-1.$ Let $T_k^d$ be the moduli space of hypersurfaces $[F]$ in ${\mathbb{P}}_k^n$ 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 $2\leq d\leq \frac{l+1}{2},$anyirreduciblecomponent$Z$of$T^d_k$satisfies$Z=T^1_k$or$\dim Z<\dim T^1_k.$