The closure of a linear space in a product of lines

By Federico Ardila, Adam Boocher. 2013

54 citations recorded by the source corpus.

Given a linear space L in affine space $${\mathbb {A}}^n$$An, we study its closure $$\widetilde{L}$$L~ in the product of projective lines $$({\mathbb {P}}^1)^n$$(P1)n. We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal $$I(\widetilde{L})$$I(L~) are all combinatorially determined by the matroid M of L. We also prove that $$I(\widetilde{L})$$I(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 $$\widetilde{L}$$L~ also gives rise to two new objects with interesting properties: the cocircuit polytope and the external activity complex of a matroid.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/76aabe75758b964928e70ddec5cdbf7723350003

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Enumeration of points, lines, planes, etc (2016, methodology): “= 0 on the image of iA : P r! Pn: The collection (ac)c2Cis uniquely determined by the circuit C, up to a common multiple. A deﬁning set of multi-homogeneous equations of YA is explicitly described in [AB16]: YA = ( X c2C aczc Y d2Cnc wd = 0; Cis a circuit of M ) (P1)n: This can be used to show tha”