Variants of the Kakeya problem over an algebraically closed field

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 $R\cup \{g\ne 0\},$ 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.

Final version, comments of the referee incorporated

References in corpus (1)

• Kakeya sets over non-archimedean local rings

Cited by in corpus (1)

• An algebraic geometry version of the Kakeya problem