Smoothness in pencils of hypersurfaces over finite fields

TL;DR

This paper investigates the properties of pencils of hypersurfaces over finite fields where all members defined over the field are smooth, exploring geometric and algebraic implications.

Contribution

It introduces a new study of smooth hypersurface pencils over finite fields, focusing on their structure and properties.

Findings

Characterization of smooth hypersurface pencils over finite fields

Conditions for all members of the pencil to be smooth

Insights into the geometric structure of these pencils

Abstract

We study pencils of hypersurfaces over finite fields $\mathbb{F}_q$ such that each of the $q+1$ members defined over $\mathbb{F}_q$ is smooth.