Partial geometries in finite affine spaces

1978 ◽  
Vol 158 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Joseph A. Thas
Keyword(s):  
Affine Spaces ◽  
Finite Affine ◽  
Partial Geometries

Groups Acting on Finite Affine Spaces

1983 ◽  
Vol s2-28 (1) ◽  
pp. 27-30
Author(s):  
Brian Mortimer
Keyword(s):  
Affine Spaces ◽  
Finite Affine

scholarly journals On chromatic indices of finite affine spaces

2018 ◽  
Vol 16 (1) ◽  
pp. 67-79
Author(s):  
Gabriela Araujo-Pardo ◽  
György Kiss ◽  
Christian Rubio-Montiel ◽  
Adrián Vázquez-Ávila
Keyword(s):  
Affine Spaces ◽  
Finite Affine

scholarly journals Kakeya sets in finite affine spaces

2011 ◽  
Vol 118 (1) ◽  
pp. 228-230
Author(s):  
Antonio Maschietti
Keyword(s):  
Affine Spaces ◽  
Finite Affine

The Erdős-Ko-Rado theorem for finite affine spaces

2016 ◽  
Vol 65 (3) ◽  
pp. 593-599 ◽  
Author(s):  
Jun Guo ◽  
Qiuli Xu
Keyword(s):  
Affine Spaces ◽  
Finite Affine

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

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.

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