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

TL;DR

This paper studies the structure of the moduli space of hypersurfaces in projective space with large degree, focusing on those with singular loci containing subschemes of specified dimension and degree, revealing a dichotomy in the components.

Contribution

It characterizes the irreducible components of the moduli space of hypersurfaces with prescribed singular subschemes, showing that only the component corresponding to degree one dominates in dimension.

Findings

Irreducible components with larger degree subschemes are of smaller dimension.

The main component corresponds to hypersurfaces singular along a linear subscheme.

For large degree, the moduli space's structure simplifies to a dominant component and smaller ones.

Abstract

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 $\dim Z<\dim T^1_k.$