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The closure of a linear space in a product of lines

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Abstract

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.

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Notes

  1. Sometimes the dual choice is made: One may also associate with L the dual matroid of rank d, whose bases are the d-subsets \(S \subset [n]\) such that \(H_S\) contains \(\pi (L)\). These two choices are equivalent, and we have chosen the one that is more convenient for us.

  2. Different conventions are used by different authors, and sometimes the names of these two polynomials are reversed. We follow [6].

  3. Sometimes the dual convention is chosen, and the matroid of L is defined to be the dual matroid \(M(L)^*\).

  4. When the choice of the order \(<\) is clear, we will omit the subscript and write simply EA(B) and EP(B).

  5. It is more common to demand that \(\sum _{x \le z \le y} \mu (x,z) = 0\) for all \(x<y\); these two conditions are equivalent.

  6. We reverse it because the initial term \({{\mathrm{in}}}_<f\) is the largest monomial of f, while basis activities are usually defined in terms of the smallest elements of circuits and cocircuits.

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Acknowledgments

We would like to thank Lauren Williams for organizing an open problem session at UC Berkeley in the Spring of 2013, where this joint project was born. The first author would also like to thank Lauren and UC Berkeley for their hospitality during the academic year 2012–2013, when part of this work was carried out. This was done while the second author was a PhD thesis student and he is grateful to David Eisenbud, Binglin Li, and Bernd Sturmfels for helpful conversations concerning this project. Finally we thank the referees for their suggestions.

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Authors and Affiliations

  1. San Francisco State University, San Francisco, CA, USA

    Federico Ardila

  2. Universidad de Los Andes, Bogotá, Colombia

    Federico Ardila

  3. University of California, Berkeley, CA

    Adam Boocher

Authors
  1. Federico Ardila
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  2. Adam Boocher
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Correspondence to Adam Boocher.

Additional information

Ardila was partially supported by the US National Science Foundation CAREER Award DMS-0956178 and the SFSU-Colombia Combinatorics Initiative. Boocher was partially supported by an NSF Graduate Research Fellowship.

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Ardila, F., Boocher, A. The closure of a linear space in a product of lines. J Algebr Comb 43, 199–235 (2016). https://doi.org/10.1007/s10801-015-0634-x

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