On construction of k-regular maps to Grassmannians via algebras of socle dimension two

TL;DR

This paper constructs new examples of k-regular maps from complex affine spaces to Grassmannians using algebraic geometry, particularly Hilbert schemes, extending classical notions and proving irreducibility results.

Contribution

It introduces novel k-regular maps for both classical and higher-dimensional Grassmannians, utilizing algebraic geometry techniques and analyzing Hilbert scheme loci.

Findings

New k-regular maps for  7; au  and au  cases.

Proved irreducibility of the punctual Hilbert scheme of 11 points on a threefold.

Established dimension counts for loci in the Hilbert scheme.

Abstract

A continuous map $\mathbb{C}^n\to Gr(\tau, N)$ is $k$-regular if the $\tau$-dimensional subspaces corresponding to images of any $k$ distinct points span a $\tau k$-dimensional space. For $\tau = 1$ this essentially recovers the classical notion of a $k$-regular map $\mathbb{C}^n\to \mathbb{C}^N$. We provide new examples of $k$-regular maps, both in the classical setting $\tau = 1$ and for $\tau\geq 2$, where these are the first examples known. Our methods come from algebraic geometry, following and generalizing Buczy\'{n}ski-Januszkiewicz-Jelisiejew-Micha{\l}ek. The key and highly nontrivial part of the argument is proving that certain loci of the Hilbert scheme of points have expected dimension. As an important side result, we prove irreducibility of the punctual Hilbert scheme of $k$ points on a threefold, for $k\leq 11$.