The close relation between border and Pommaret marked bases

Given a finite order ideal O in the polynomial ring K[x1, … , xn] over a field K, let ∂O be the border of O and PO the Pommaret basis of the ideal generated by the terms outside O. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among ∂O-marked sets (resp. bases) and PO-marked sets (resp. bases). We prove that a ∂O-marked set B is a marked basis if and only if the PO-marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing ∂O-marked bases and PO-marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.