Computing subschemes of the border basis scheme

TL;DR

This paper develops explicit algorithms to identify and compute loci within border basis schemes that correspond to 0-dimensional schemes with specific algebraic properties, enhancing the understanding of their structure.

Contribution

It introduces effective algorithms for determining loci with properties like Gorenstein and Cayley-Bacharach within border basis schemes, with concrete examples.

Findings

Algorithms for computing defining ideals of loci with special properties.

Explicit descriptions of loci for Gorenstein, Cayley-Bacharach, and complete intersection schemes.

Illustrative examples demonstrating the effectiveness of the methods.

Abstract

A good way of parametrizing 0-dimensional schemes in an affine space $\mathbb{A}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $\mu$, they provide an open covering of the Hilbert scheme ${\rm Hilb}^\mu(\mathbb{A}^n_K)$ and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent 0-dimensional $K$-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley-Bacharach, and strict Cayley-Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by non-trivial, concrete examples.