Computing subschemes of the border basis scheme

Author:

Kreuzer Martin1,Long Le Ngoc12,Robbiano Lorenzo3

Affiliation:

1. Fakultät für Informatik und Mathematik, Universität Passau, D-94030 Passau, Germany

2. Department of Mathematics, University of Education – Hue University, 34 Le Loi, Hue, Vietnam

3. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova, Italy

Abstract

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] 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 zero-dimensional [Formula: see text]-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 nontrivial, concrete examples.

Funder

Vietnam National Foundation for Science and Technology Development

Publisher

World Scientific Pub Co Pte Lt

Subject

General Mathematics

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