The close relation between border and Pommaret marked bases

• Cristina Bertone ^[1] ; Francesca Cioffi ^[2]
  1. [1] Dipartimento di Matematica dell’Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy
  2. [2] Dipartimento di Matematica e Applicazioni dell’Università di Napoli Federico II, via Cintia, 80126, Napoli, Italy
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 2, 2022, págs. 181-201
• Idioma: inglés
• DOI: 10.1007/s13348-021-00313-w
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• Resumen

  • Given a finite order ideal {\mathcal {O}} in the polynomial ring K[x_1,\ldots , x_n] over a field K, let \partial {\mathcal {O}} be the border of {\mathcal {O}} and {\mathcal {P}}_{\mathcal {O}} the Pommaret basis of the ideal generated by the terms outside {\mathcal {O}}. In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among \partial {\mathcal {O}}-marked sets (resp. bases) and {\mathcal {P}}_{\mathcal {O}}-marked sets (resp. bases). We prove that a \partial {\mathcal {O}}-marked set B is a marked basis if and only if the {\mathcal {P}}_{\mathcal {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}}-marked bases and {\mathcal {P}}_{\mathcal {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.