Geodesic
The closure of a linear space in a product of lines
Journal of Algebraic Combinatorics, Tome 43 (2016) no. 1, pp. 199-235.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Given a linear space $L$ in affine space ${\mathbb {A}}^n$, we study its closure $\widetilde{L}$ in the product of projective lines $({\mathbb {P}}^1)^n$. We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal $I(\widetilde{L})$ are all combinatorially determined by the matroid $M$ of $L$. We also prove that $I(\widetilde{L})$ and all of its initial ideals are Cohen-Macaulay with the same Betti numbers, and can be used to compute the $h$-vector of $M$. This variety $\widetilde{L}$ also gives rise to two new objects with interesting properties: the cocircuit polytope and the external activity complex of a matroid.
Classification : 05B35, 13P10, 52B40
Keywords: Betti numbers, universal Gröbner bases, matroids, state polytopes, Stanley-Reisner ideals 
@article{JAC_2016__43_1_a1,
     author = {Ardila, Federico and Boocher, Adam},
     title = {The closure of a linear space in a product of lines},
     journal = {Journal of Algebraic Combinatorics},
     pages = {199--235},
     publisher = {mathdoc},
     volume = {43},
     number = {1},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2016__43_1_a1/}
}
TY  - JOUR
AU  - Ardila, Federico
AU  - Boocher, Adam
TI  - The closure of a linear space in a product of lines
JO  - Journal of Algebraic Combinatorics
PY  - 2016
SP  - 199
EP  - 235
VL  - 43
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JAC_2016__43_1_a1/
LA  - en
ID  - JAC_2016__43_1_a1
ER  - 
%0 Journal Article
%A Ardila, Federico
%A Boocher, Adam
%T The closure of a linear space in a product of lines
%J Journal of Algebraic Combinatorics
%D 2016
%P 199-235
%V 43
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JAC_2016__43_1_a1/
%G en
%F JAC_2016__43_1_a1
Ardila, Federico; Boocher, Adam. The closure of a linear space in a product of lines. Journal of Algebraic Combinatorics, Tome 43 (2016) no. 1, pp. 199-235. http://geodesic.mathdoc.fr/item/JAC_2016__43_1_a1/