  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.

Commutative Algebra;Algebraic Geometry