The moduli space of hypersurfaces whose singular locus has high dimension

Abstract

Let $k$ be an algebraically closed field and let $b$ and $n$ be integers with $n\ge 3$ and $1\le b\le n-1.$ Consider the moduli space $X$ of hypersurfaces in ${\mathbb{P}}_k^n$ 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.

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