Constructions of \(k\)-regular maps using finite local schemes (Q2418887)

Summary: 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 \(k \leq 9\), and we provide explicit examples for \(k \leq 5\). We also provide upper bounds for arbitrary \(m\) and \(k\).