Re-Embeddings of Special Border Basis Schemes
Martin Kreuzer, Lorenzo Robbiano
TL;DR
This work advances the program of re-embedding border basis schemes into low-dimensional affine spaces by developing and applying $Z$-separating re-embeddings across several special families: MaxDeg, L-shape, simplicial, and planar border basis schemes. It establishes when such schemes admit affine-cell realizations, often via nonnegative arrow gradings and elimination of non-exposed indeterminates, and it provides explicit constructions (including the Unimodular Matrix Problem) to achieve polynomial-coordinate realizations in notable cases like the L-shape. The results yield concrete, computable embeddings, exemplified by proving that the L-shape border basis scheme is isomorphic to an affine space $\mathbb{A}^{10}$ and by detailing optimal re-embeddings for simplicial and planar cases, with several conjectures guiding the limits of these methods. Collectively, the paper deepens understanding of when border basis schemes are affine cells and offers practical algorithms for their computational handling, contributing to the broader study of moduli spaces of zero-dimensional ideals.
Abstract
Border basis schemes are open subschemes of the Hilbert scheme of $μ$ points in an affine space $\mathbb{A}^n$. They have easily describable systems of generators of their vanishing ideals for a natural embedding into a large affine space $\mathbb{A}^{μν}$. Here we bring together several techniques for re-embedding affine schemes into lower dimensional spaces which we developed in the last years. We study their efficacy for some special types of border basis schemes such as MaxDeg border basis schemes, L-shape and simplicial border basis schemes, as well as planar border basis schemes. A particular care is taken to make these re-embeddings efficiently computable and to check when we actually get an isomorphism with $\mathbb{A}^{nμ}$, i.e., when the border basis scheme is an affine cell.