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 $R\cup\{g\neq 0\},$ 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.Comment: Final version, comments of the referee incorporate

A finite version of the Kakeya problem

“…The aim of this article is to reformulate the Kakeya problem in a far more general setting. There are a couple of recent articles by Slavov in which he formulates the Kakeya problem in an algebraic geometric setting, see [6] and [7]. Here we consider any arbitrary finite set of lines in an affine space over any fixed field K. We prove lower bounds on the size of S that depend on I(D), the ideal generated by the homogeneous polynomials of K[X 1 , .…”