AbstractGiven a finite order ideal $${\mathcal {O}}$$
O
in the polynomial ring $$K[x_1,\ldots , x_n]$$
K
[
x
1
,
…
,
x
n
]
over a field K, let $$\partial {\mathcal {O}}$$
∂
O
be the border of $${\mathcal {O}}$$
O
and $${\mathcal {P}}_{\mathcal {O}}$$
P
O
the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$
O
. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$
∂
O
-marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$
P
O
-marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$
∂
O
-marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$
P
O
-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 $$\partial {\mathcal {O}}$$
∂
O
-marked bases and $${\mathcal {P}}_{\mathcal {O}}$$
P
O
-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.
Local rigidity of certain solvable group actions on tori
