Variants of the Kakeya problem over an algebraically closed field

TL;DR

This paper investigates algebraic geometry variants of the Kakeya problem, classifying minimal constructible subsets containing lines in all directions and analyzing subvarieties with maximal directional line sets.

Contribution

It provides a classification of minimal Kakeya-type subsets in affine 3-space and studies subvarieties with maximal directional line sets in an algebraic setting.

Findings

Classified smallest Kakeya-type subsets in -space.

Analyzed subvarieties with maximal line directions.

Extended Kakeya problem to algebraic geometry context.

Abstract

First, we study constructible subsets of $\A^n_k$ which contain a line in any direction. We classify the smallest such subsets in $\A^3$ of the type $R\cup\{g\neq 0\},$ where $g\in k[x_1,...,x_n]$ is irreducible of degree $d$, and $R\subset V(g)$ is closed. Next, we study subvarieties $X\subset\A^N$ for which the set of directions of lines contined in $X$ has the maximal possible dimension. These are variants of the Kakeya problem in an algebraic geometry context.