Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The moduli space of hypersurfaces whose singular locus has high dimension

Kaloyan Slavov

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.

The moduli space of hypersurfaces whose singular locus has high dimension

Abstract

Let be an algebraically closed field and let and be integers with and Consider the moduli space of hypersurfaces in of fixed degree whose singular locus is at least -dimensional. We prove that for large , has a unique irreducible component of maximal dimension, consisting of the hypersurfaces singular along a linear -dimensional subspace of . The proof will involve a probabilistic counting argument over finite fields.

Paper Structure

This paper contains 21 sections, 32 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The incidence correspondence and the restricted Hilbert scheme
  4. The incidence correspondence
  5. The restricted Hilbert scheme
  6. Specialization arguments
  7. Specialization of a closed subscheme to a union of linear subspaces
  8. An upper bound on the dimension of the space of $F$ such that $C\subset V(F)_{\mathop{\mathrm{sing}}\nolimits}$, for a fixed $C$ of small degree
  9. An upper bound on the dimension of the space of $F$ such that $C\subset V(F)$, for a fixed $C$ of small degree
  10. The case of small degree $d$
  11. The component corresponding to $d=1$
  12. The case $d\leq B$ (small degree)
  13. Preparations for the computation of the second largest component
  14. The case of large degree $d$, when $k=\overline{\mathbb{F}_p}$
  15. Reduction to a problem over finite fields
  16. ...and 6 more sections

Key Result

Theorem \oldthetheorem

There exists an effectively computable integer $l_0=l_0(n,b,p),$ such that for all $l\geq l_0,$$X^1$ is the unique irreducible component of $X$ of maximal dimension.

Theorems & Definitions (70)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Chow's finiteness theorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • ...and 60 more