Constructions of $k$-regular maps using finite local schemes

Abstract

A continuous map ${\mathbb{R}}^m\to {\mathbb{R}}^N$ or ${\mathbb{C}}^m\to {\mathbb{C}}^N$ is called $k$-regular if the images of any $k$ points are linearly independent. Given integers $m$ and $k$ a problem going back to Chebyshev and Borsuk is to determine the minimal value of $N$ for which such maps exist. The methods of algebraic topology provide lower bounds for $N$, but there are very few results on the existence of such maps for particular values $m$ and $k$. Using methods of algebraic geometry we construct $k$-regular maps. We relate the upper bounds on $N$ with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for $k\le 9$, and we provide explicit examples for $k\le 5$. We also provide upper bounds for arbitrary $m$ and $k$.