On the equations defining some Hilbert schemes

TL;DR

This paper derives explicit equations for specific Hilbert schemes, enhancing understanding of their geometry and providing tools for further computational and theoretical exploration.

Contribution

It provides explicit equations for the Hilbert scheme of two points on the projective plane and for non-planar clusters in the punctual Hilbert scheme, addressing a question by Sturmfels.

Findings

Explicit equations for the Hilbert scheme of two points on the projective plane.

Explicit equations for non-planar clusters in the punctual Hilbert scheme of length four.

Equations are governed by SL_3 representation theory and computer algebra.

Abstract

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme of two points on the projective plane and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme of clusters of length four on affine three-space with support at the origin. We find explicit equations in natural projective, respectively affine embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL_3, our arguments also rely on computer algebra.