Riemann–Roch theory for graph orientations

Backman, Spencer

doi:10.1016/j.aim.2017.01.005

Cited by 33 publications

(66 citation statements)

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“…Then Section 2 establishes the main results for orientations and their behavior under edge-contractions, proving Theorem 1.1.1. Our work here has been influenced by [21] and [6], which study the interplay between orientations and the divisors they define. In Section 3 we treat compactified Jacobians and prove Theorem 1.1.2.…”

Section: Introductionmentioning

confidence: 99%

Combinatorics of compactified universal Jacobians

Caporaso

Christ

2019

Advances in Mathematics

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We show that the combinatorial structure of the compactified universal Jacobians over M g in degrees g − 1 and g is governed by orientations on stable graphs. In particular, for a stable curve we exhibit graded stratifications of the compactified Jacobians in terms of totally cyclic, respectively rooted, orientations on its dual graph. We prove functoriality under edge-contraction of the posets of totally cyclic and rooted orientations on stable graphs. 9 9This holds in degrees g − 1 and g and can be easily extended to degree g − 2 (by taking the residual of the degree g case).The theorem gives the sought-for combinatorial presentation of the compactified Jacobian of a curve, and of the compactified universal Jacobian over M g . The next question now is to provide the tropical/non-Archimedian version of the theorem, starting from the fact that the left-bottom corner of the diagram should be occupied by the moduli space of tropical curves, M g trop , while the right side should be the same, up to isomorphism. This will involve constructing skeleta of P d X and P d g as moduli spaces of suitable polyhedral objects. This research direction relates to results of [8], where

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Section: Introductionmentioning

confidence: 99%

Combinatorics of compactified universal Jacobians

Caporaso

Christ

2019

Advances in Mathematics

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“…12 The results of [Bac14] can be used to prove that the inverse of the natural map B(G) → P ic g (G) is efficiently computable. See [BS13] for an explanation of how such a bijection can be used to find random spanning trees.…”

Section: G) From This We Get Amentioning

confidence: 99%

“…There is an efficient (polynomial-time) algorithm for deciding whether or not a given divisor on a graph is a break divisor; see [Bac14].…”

Section: The Bernardi Map Is Bijectivementioning

confidence: 99%

The Bernardi Process and Torsor Structures on Spanning Trees

Baker

Wang

2017

International Mathematics Research Notices

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Abstract. Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to Olivier Bernardi, we associate to any pair (v, e) consisting of a vertex v and an edge e adjacent to v a bijection β (v,e) between spanning trees of G and elements of the set Pic g (G) of degree g divisor classes on G, where g is the genus of G in the sense of Baker-Norine. We give a new proof that the map β (v,e) is bijective by explicitly constructing an inverse. Using the natural action of the Picard group Pic 0 (G) on Pic g (G), we show that the Bernardi bijection β (v,e) gives rise to a simply transitive action β v of Pic 0 (G) on the set of spanning trees which does not depend on the choice of e. A plane graph has a natural ribbon structure (coming from the counterclockwise orientation of the plane), and in this case we show that β v is independent of v as well. Thus for plane graphs, the set of spanning trees is naturally a torsor for the Picard group. Conversely, we show that if β v is independent of v then G together with its ribbon structure is planar. We also show that the natural action of Pic 0 (G) on spanning trees of a plane graph is compatible with planar duality.These findings are formally quite similar to results of Holroyd et al. and ChanChurch-Grochow, who used rotor-routing to construct an action r v of Pic 0 (G) on the spanning trees of a ribbon graph G, which they show is independent of v if and only if G is planar. It is therefore natural to ask how the two constructions are related. We prove that β v = r v for all vertices v of G when G is a planar ribbon graph, i.e. the two torsor structures (Bernardi and rotor-routing) on the set of spanning trees coincide. In particular, it follows that the rotor-routing torsor is compatible with planar duality. We conjecture that for every non-planar ribbon graph G, there exists a vertex v with β v = r v .We thank Spencer Backman, Melody Chan, and Dan Margalit for helpful discussions and feedback on an earlier draft. We also thank Chi Ho Yuen for catching a sign error in Theorem 6.1, and the anonymous referees for their extraordinarily careful proofreading and numerous useful suggestions. This work began at the American Insitute of Mathematics workshop "Generalizations of chip firing and the critical group" in July 2013, and we would like to thank AIM as well as the organizers of that conference (L. Levine, J. Martin, D. Perkinson, and J. Propp) for providing a stimulating environment.

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“…, v r are the columns of A. 4 Remark 3.1.3. When M = M (G) is a graphic matroid, it is usually more convenient to take A to be the full adjacency matrix of G, rather than a modified adjacency matrix with one row removed, when defining the corresponding zonotope.…”

Section: 1mentioning

confidence: 99%

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

Backman

Baker

Yuen

2019

Forum of Mathematics, Sigma

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Let M be a regular matroid. The Jacobian group Jac(M ) of M is a finite abelian group whose cardinality is equal to the number of bases of M . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) Jac(G) of a graph G (in which case bases of the corresponding regular matroid are spanning trees of G).There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph Jac(G) and spanning trees. However, most of the known bijections use vertices of G in some essential way and are inherently "non-matroidal". In this paper, we construct a family of explicit and easy-todescribe bijections between the Jacobian group of a regular matroid M and bases of M , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of M admits a canonical simply transitive action on the set G(M ) of circuit-cocircuit reversal classes of M , and then define a family of combinatorial bijections β σ,σ * between G(M ) and bases of M . (Here σ (resp. σ * ) is an acyclic signature of the set of circuits (resp. cocircuits) of M .) We then give a geometric interpretation of each such map β = β σ,σ * in terms of zonotopal subdivisions which is used to verify that β is indeed a bijection.Finally, we give a combinatorial interpretation of lattice points in the zonotope Z; by passing to dilations we obtain a new derivation of Stanley's formula linking the Ehrhart polynomial of Z to the Tutte polynomial of M . arXiv:1701.01051v3 [math.CO] 23 Apr 20191 If e and e * are dual edges of G and G * , respectively, then given an orientation for e * we orient e by rotating the orientation of e * clockwise locally near the crossing of e and e * .

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Riemann–Roch theory for graph orientations

Cited by 33 publications

References 35 publications

Combinatorics of compactified universal Jacobians

Combinatorics of compactified universal Jacobians

The Bernardi Process and Torsor Structures on Spanning Trees

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory

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