Variants of the Kakeya problem over an algebraically closed field

Abstract

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