Density of quasismooth hypersurfaces in simplicial toric varieties

TL;DR

This paper extends Poonen's formula to compute the density of quasismooth hypersurfaces in simplicial toric varieties over finite fields, with applications to bounding singularities.

Contribution

It generalizes Poonen's density formula from smooth to quasismooth hypersurfaces in simplicial toric varieties.

Findings

Derived a density formula for quasismooth hypersurfaces in simplicial toric varieties.

Analyzed bounds on singularities and their schemes in hypersurfaces.

Extended existing results to a broader class of algebraic varieties.

Abstract

This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B. Poonen for smooth projective varieties. As an application, we further analyze the density of hypersurfaces with bounds on their number of singularities and on the length of their singular schemes.