Existence of pencils with nonblocking hypersurfaces

TL;DR

This paper proves the existence of a family of hypersurfaces over finite fields where each member avoids blocking lines, expanding understanding of geometric configurations in finite projective spaces.

Contribution

It establishes the existence of hypersurface pencils over finite fields with all members nonblocking, a novel result in finite algebraic geometry.

Findings

Existence of such hypersurface pencils for any degree and dimension.

All members of the pencil are nonblocking with respect to finite field lines.

Advances understanding of geometric configurations over finite fields.

Abstract

We prove that there is a pencil of hypersurfaces in $\mathbb{P}^n$ of any given degree over a finite field $\mathbb{F}_q$ such that every $\mathbb{F}_q$-member of the pencil is not blocking with respect to $\mathbb{F}_q$-lines.