Transverse-freeness in finite geometries

TL;DR

This paper investigates the properties of projective hypersurfaces over finite fields that are tangent to all members of certain low-degree families, extending previous work and exploring finite models of classical geometries.

Contribution

It extends the study of tangent hypersurfaces to higher dimensions and constructs tangent curves in finite models of inversive and hyperbolic planes.

Findings

Determines minimal degrees of tangent hypersurfaces in projective spaces.

Constructs tangent curves in finite models of classical geometries.

Extends 2D results to higher dimensions and geometries.

Abstract

We study projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in $n$-dimensional projective space that are tangent to every $k$-dimensional subspace, for some value of $n$ and $k$. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces we construct curves tangent to each of the lowest degree curves defined over the base field.