Toric vector bundles and parliaments of polytopes

Rocco, Sandra Di; Jabbusch, Kelly; Smith, Gregory G.

doi:10.1090/tran/7201

Cited by 19 publications

(32 citation statements)

References 14 publications

(17 reference statements)

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“…Restricting equivariant vector bundles to invariant curves. We recall the recipe of restricting an equivariant vector bundle to an invariant curve from [HMP,Section 5], [DJS,Section 5]. Let X = X(∆) be a nonsingular complete complex toric variety under the action of the torus T , determined by the fan ∆ in the lattice N ∼ = Z n .…”

Section: 2mentioning

confidence: 99%

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Seshadri constants of equivariant vector bundles on toric varieties

Dasgupta¹,

Khan²,

Subramaniam³

2021

Preprint

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We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.

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Section: 2mentioning

confidence: 99%

“…We would like to mention that there are examples of equivariant ample vector bundles which do not satisfy the assumption 5.3. One such example can be found in [DJS,Example 5.5].…”

Section: Seshadri Constants Of Equivariant Vector Bundles On Hirzebru...mentioning

confidence: 99%

Seshadri constants of equivariant vector bundles on toric varieties

Dasgupta¹,

Khan²,

Subramaniam³

2021

Preprint

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“…We denote the matroid M(A v ) by M(E). This matroid is introduced and used in [DRJS18] to define the notion of parliament of polytopes of E. We point out that in [DRJS18], the subspace arrangement used is the Klyachko arrangement. Proposition 5.13 shows that it coincides with A v .…”

Section: Free and Projectivementioning

confidence: 99%

“…In Proposition 6.14, using the criteria in [DRJS18], we observe that if all the piecewise linear functions in the image of v are convex then E is globally generated.…”

Section: Introductionmentioning

confidence: 99%

“…In [DRJS18] the authors introduce a collection of polytopes associated to a toric vector bundle E called the parliament of polytopes of E. The polytopes in the parliament are indexed by the ground set of a certain representable matroid M(E) (living in E). They give a criterion for global generation of E in terms of its parliament We can immediately recover the matroid and parliament of polytopes of E from its prevaluation (Proposition 6.13).…”

Section: Introductionmentioning

confidence: 99%

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Toric flat families, valuations, and tropical geometry over the semifield of piecewise linear functions

Kaveh¹,

Manon²

2019

Preprint

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We initiate the algebro-geometric study of tropical geometry over the idemopotent semifield of piecewise linear functions. One of our main results shows that points on the tropical variety of a linear ideal over this semifield correspond to toric vector bundles. We introduce the notion of a valuation with values in the semifield of piecewise linear functions and we describe Khovanskii bases in this context. Far extending the Klyachko classification of toric vector bundles, we show that torus equivariant families over toric varieties are classified by such valuations. Finally, we see that the Gross-Hacking-Keel-Kontsevich toric degenerations of cluster varieties fit into our picture as a family over the toric scheme of the Fock-Goncharov fan.

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Toric principal bundles, piecewise linear maps and Tits buildings

Kaveh

Manon

2022

Math. Z.

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Recent work [DJS18, KM22, KMT22, KM23] has produced new ways to encode the data of a torus equivariant vector bundle over a toric variety by certain representable matroid(s) labeled by polyhedral data. In this paper we show that this data makes sense for non-representable matroids as well. We call the resulting combinatorial objects toric matorid bundles. Alternatively, they can also be called tropical vector bundles. We define equivariant K-theory and characteristic classes of these bundles. As a particular case, we show that any matroid comes with tautological toric matroid bundles over the permutahedral toric variety and the corresponding equivariant K-classes and Chern classes agree with those in the recent work [BEST23] on matroid invariants. Moreover, in analogy with toric vector bundles, we define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for toric matroid bundles. Finally, we study the splitting of toric matroid bundles and, in particular, an analogue of Grothendieck's theorem on splitting of vector bundles on P 1 . Contents 19 7. Tautological matroid bundles on the permutahedral variety 23 8. Matroid extensions and toric matroid bundles 27 9. Splitting of toric matroid bundles and ampleness 29 References 32This is a preliminary version, comments are welcomed.

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Toric vector bundles and parliaments of polytopes

Cited by 19 publications

References 14 publications

Seshadri constants of equivariant vector bundles on toric varieties

Seshadri constants of equivariant vector bundles on toric varieties

Toric flat families, valuations, and tropical geometry over the semifield of piecewise linear functions

Toric principal bundles, piecewise linear maps and Tits buildings

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