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

### Jarosław Buczyński

University of Warsaw and Polish Academy of Sciences, Warsaw, Poland### Tadeusz Januszkiewicz

Polish Academy of Sciences, Warsaw, Poland### Joachim Jelisiejew

University of Warsaw, Poland### Mateusz Michałek

Freie Universität Berlin, Germany, and Polish Academy of Sciences, Warsaw, Poland

## Abstract

A continuous map $R_{m}→R_{N}$ or $C_{m}→C_{N}$ 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 $k≤9$, and we provide explicit examples for $k≤5$. We also provide upper bounds for arbitrary $m$ and $k$.

## Cite this article

Jarosław Buczyński, Tadeusz Januszkiewicz, Joachim Jelisiejew, Mateusz Michałek, Constructions of $k$-regular maps using finite local schemes. J. Eur. Math. Soc. 21 (2019), no. 6, pp. 1775–1808

DOI 10.4171/JEMS/873