Tautological classes of matroids

Berget, Andrew; Eur, Christopher; Spink, Hunter; Tseng, Dennis

doi:10.48550/arxiv.2103.08021

Cited by 4 publications

(10 citation statements)

References 44 publications

(79 reference statements)

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“…Speyer constructed an invariant of a matroid by associating to it the K -theory class of the structure sheaf of an associated torus orbit closure in a Grassmannian [45]; its properties were studied further by Fink and Speyer [16]. The Chow and K -theory of matroids are connected by a beautiful recent paper of Berget, Eur, Spink, and Tseng [5], who connect wonderful models of matroids to equivariant vector bundles on toric varieties. In fact, this construction plays a crucial role in our study.…”

Section: Previous Workmentioning

confidence: 99%

“…We work over an algebraically closed field of characteristic 0 and construct the virtual classes of matroids. The section involves three ingredients: the tautological vector bundle cutting out the wonderful model of a matroid [5], the Grothendieck-Riemann-Roch theorem [10], and the functoriality of the virtual fundamental class [33]. They will be combined to obtain a combinatorial formula for the virtual fundamental class of a wonderful model for a complex arrangement complement.…”

Section: Virtual Fundamental Classes Of Wonderful Modelsmentioning

confidence: 99%

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Gromov–Witten theory and invariants of matroids

Ranganathan

Usatine

2022

Sel. Math. New Ser.

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We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids.

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Section: Previous Workmentioning

confidence: 99%

Section: Virtual Fundamental Classes Of Wonderful Modelsmentioning

confidence: 99%

Gromov–Witten theory and invariants of matroids

Ranganathan

Usatine

2022

Sel. Math. New Ser.

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show abstract

“…Speyer constructed an invariant of a matroid by associating to it the K-theory class of the structure sheaf of an associated torus orbit closure in a Grassmannian [43]; its properties were studied further by Fink and Speyer [15]. The Chow and K-theory of matroids are connected by a beautiful recent paper of Berget, Eur, Spink, and Tseng [5], who connect wonderful models of matroids to equivariant vector bundles on toric varieties. In fact, this construction plays a crucial role in our study.…”

Section: Theorem C (Tropical Virtual Class)mentioning

confidence: 99%

“…We work over an algebraically closed field of characteristic 0 and our logarithmic structures in this section will be fine but not necessarily saturated. The section involves three ingredients: the tautological vector bundle cutting out the wonderful model of a matroid [5], the virtual Grothendieck-Riemann-Roch theorem [10], and the functoriality of the virtual fundamental class [31]. They will be combined to obtain a combinatorial formula for the virtual fundamental class of a wonderful model for a complex arrangement complement.…”

Section: Virtual Fundamental Classes Of Wonderful Modelsmentioning

confidence: 99%

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Gromov-Witten theory and invariants of matroids

Ranganathan¹,

Usatine²

2021

Preprint

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We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov-Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement's wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov-Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a welldefined Gromov-Witten theory of non-realizable matroids.DHRUV RANGANATHAN, DEPARTMENT

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Stellahedral geometry of matroids

Eur¹,

Huh²,

Larson³

2022

Preprint

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We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety, and show that valuative, homological, and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shaprio-Smirnov-Vaintrob on Postnikov-Shaprio algebras, and calculate the Chern-Schwartz-MacPherson classes of matroid Schubert cells. The central construction is the "augmented tautological classes of matroids," modeled after certain vector bundles on the stellahedral toric variety. CONTENTS 1. Introduction 1 2. Torus-equivariant geometry preliminaries 3. Stellahedral varieties 4. Augmented tautological bundles and classes 5. Augmented wonderful varieties and Bergman classes 6. Exceptional isomorphisms 7. Valuative group, homology, and the intersection pairing 8. Numerical properties 9. Chern-Schwartz-MacPherson classes Appendix A. Polytope algebras and K-rings of toric varieties References

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Tautological classes of matroids

Cited by 4 publications

References 44 publications

Gromov–Witten theory and invariants of matroids

Gromov–Witten theory and invariants of matroids

Gromov-Witten theory and invariants of matroids

Stellahedral geometry of matroids

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