Hypersurfaces with defect

TL;DR

This paper investigates the properties of hypersurfaces with defect, establishing bounds on their Tjurina number in characteristic zero and positive characteristic, and analyzing their density over finite fields.

Contribution

It provides new bounds on the Tjurina number for hypersurfaces with defect and explores their density over finite fields, extending understanding of their geometric and arithmetic properties.

Findings

Tjurina number of hypersurfaces with defect is large in characteristic 0

Similar bounds hold in positive characteristic for mildly singular hypersurfaces

Lower bounds on the density of defect-free hypersurfaces over finite fields

Abstract

A projective hypersurface $X \subseteq \mathbb P^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$ for some $i \in \{n, \dots, 2n-2\}$ in a suitable cohomology theory. This occurs for example when $X \subseteq \mathbb P^4$ is not $\mathbb Q$-factorial. We show that in characteristic 0, the Tjurina number of hypersurfaces with defect is large. For $X$ with mild singularities, there is a similar result in positive characteristic. As an application, we obtain a lower bound on the asymptotic density of hypersurfaces without defect over a finite field.