We introduce the conormal fan of a matroid M, which is a Lagrangian analog of the Bergman fan of M. We use the conormal fan to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M. This allows us to express the h-vector of the broken circuit complex of M in terms of the intersection theory of the conormal fan of M. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. The Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of M, when combined with the Hodge-Riemann relations for the conormal fan of M, implies Brylawski's and Dawson's conjectures that the h-vectors of the broken circuit complex and the independence complex of M are log-concave sequences.

Lagrangian geometry of matroids

We introduce the conormal fan of a matroid <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is a Lagrangian analog of the Bergman fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This allows us to express the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vector of the broken circuit complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the intersection theory of the conormal fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, when combined with the Hodge–Riemann relations for the conormal fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, implies Brylawski’s and Dawson’s conjectures that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vectors of the broken circuit complex and the independence complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are log-concave sequences. 

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $${\mathbb {Z}}$$
 -graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.

/pdf/divisors-on-graphs-binomial-and-monomial-ideals-and-cellular-1v2u80bf8d.pdf

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

Betti numbers of symmetric shifted ideals

We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.

Castelnuovo-Mumford regularity up to symmetry

Codimension and projective dimension up to symmetry

Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain If the ideals are graded it is known that the codimensions grow eventually linearly Here this result is extended to chains of arbitrary symmetric ideals Moreover, the slope of the linear function is explicitly determined We conjecture that the projective dimensions also grow eventually linearly As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals As an application, this yields Cohen-Macaulayness obstructions

Codimension and Projective Dimension up to Symmetry

Flawlessness of h-vectors of broken circuit complexes

We relate finite generation of cones, monoids, and ideals in increasing chains (the local situation) to equivariant finite generation of the corresponding limit objects (the global situation). For cones and monoids there is no analogue of Noetherianity as in the case of ideals and we demonstrate this in examples. As a remedy we find local-global correspondences for finite generation. These results are derived from a more general framework that relates finite generation under closure operations to equivariant finite generation under general families of maps. We also give a new proof that non-saturated Inc-invariant chains stabilize, closing a gap in the literature.

Invariant chains in algebra and discrete geometry

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