The moduli space of hypersurfaces whose singular locus has high dimension

TL;DR

This paper investigates the structure of the moduli space of hypersurfaces in projective space with high-dimensional singular loci, showing that for large degrees, the space is dominated by hypersurfaces singular along linear subspaces.

Contribution

It establishes the uniqueness of the maximal irreducible component of the moduli space for large degrees, characterized by hypersurfaces singular along linear subspaces.

Findings

Unique maximal irreducible component identified

Hypersurfaces singular along linear subspaces dominate for large degree

Probabilistic counting over finite fields used in proof

Abstract

Let $k$ be an algebraically closed field and let $b$ and $n$ be integers with $n\geq 3$ and $1\leq b \leq n-1.$ Consider the moduli space $X$ of hypersurfaces in $\mathbb{P}^n_k$ of fixed degree $l$ whose singular locus is at least $b$-dimensional. We prove that for large $l$, $X$ has a unique irreducible component of maximal dimension, consisting of the hypersurfaces singular along a linear $b$-dimensional subspace of $\mathbb{P}^n$. The proof will involve a probabilistic counting argument over finite fields.