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November 2019 The Chow Form of a Reciprocal Linear Space
Mario Kummer, Cynthia Vinzant
Michigan Math. J. 68(4): 831-858 (November 2019). DOI: 10.1307/mmj/1571731287


A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.


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Mario Kummer. Cynthia Vinzant. "The Chow Form of a Reciprocal Linear Space." Michigan Math. J. 68 (4) 831 - 858, November 2019.


Received: 28 November 2017; Revised: 18 December 2017; Published: November 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07155051
MathSciNet: MR4029631
Digital Object Identifier: 10.1307/mmj/1571731287

Primary: 14M12
Secondary: 05B35 , 13C14 , 14M15 , 52C35

Rights: Copyright © 2019 The University of Michigan


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Vol.68 • No. 4 • November 2019
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