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

Abstract

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 $2\le d\le \frac{l+1}{2},$ any irreducible component $Z$ of $T_k^d$ satisfies $Z=T_k^1$ or $\dim Z<\dim T_k^1.$

arXiv commentThis paper has been withdrawn because it has been incorporated into a longer paper