Abstract
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.
Funder
Università degli Studi di Torino
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics
Reference31 articles.
1. Abbott, J., Bigatti, A.M., Robbiano, L.: CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it
2. Bertone, C., Cioffi, F.: On almost revlex ideals with Hilbert function of complete intersections. Ric. Mat. 69(1), 153–175 (2020). https://doi.org/10.1007/s11587-019-00453-z
3. Bertone, C., Cioffi, F., Lella, P., Roggero, M.: Upgraded methods for the effective computation of marked schemes on a strongly stable ideal. J. Symbolic Comput. 50, 263–290 (2013). https://doi.org/10.1016/j.jsc.2012.07.006
4. Bertone, C., Cioffi, F., Roggero, M.: Macaulay-like marked bases. J. Algebra Appl. 16(5), 1750100, 36 (2017). https://doi.org/10.1142/S0219498817501006
5. Bertone, C., Cioffi, F., Roggero, M.: Smoothable Gorenstein points via marked schemes and double-generic initial ideals. Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1592034
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