Geometry of hyperfields

Jun, Jaiung

doi:10.48550/arxiv.1707.09348

Cited by 9 publications

(12 citation statements)

References 20 publications

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“…There is a canonical way to impose a topology on X (k), called the fine Zariski topology. In tropical geometry, the fine Zariski topology has been used to give a homeomorphism between Berkovich analytification and a set of rational points of a scheme over some "generalized algebraic structures", for instance, see [GG14], [Lor15], and [Jun17b].…”

Section: Topological T-vector Bundlesmentioning

confidence: 99%

Vector bundles on tropical schemes

Jun,

Mincheva,

Tolliver

2020

Preprint

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We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.

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Section: Topological T-vector Bundlesmentioning

confidence: 99%

Vector bundles on tropical schemes

Jun,

Mincheva,

Tolliver

2020

Preprint

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“…That nonarchimedean seminorms can be interpreted as morphisms of hyperrings was observed Viro in [23]; also cf. [7] and [11]. In so far, the following theorem does not contain a novel mathematical fact, though its appearance in terms of ordered blueprints is new.…”

Section: Nonarchimedean Seminorms As Morphismsmentioning

confidence: 99%

“…The Kajiwara-Payne tropicalization as a rational point set. Jun observes in [11] that the Berkovich analytification of X corresponds to the hyperring morphism from R, considered as a hyperring, into the tropical hyperfield T. We transfer this approach to ordered blueprints, which allows us to recover both the analytification and the tropicalization of X = Spec R as T-rational point sets of the following scheme theoretic tropicalizations.…”

Section: Valuations As Morphismsmentioning

confidence: 99%

Tropical geometry over the tropical hyperfield

Lorscheid

2019

Preprint

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In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety.Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization. ContentsIntroduction 1 1. Ordered blueprints 8 2. Tropicalization as a base change to the tropical hyperfield 13 3. The Kajiwara-Payne tropicalization as a rational point set 18 4. The relation between the tropical hyperfield and the tropical semifield 25 5. Recovering the Giansiracusa bend 27 References 32

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“…For ease of notation the symbol ⊥ is to be understood in context as either ⊥ s or ⊥ w . [13], the second author proves that certain topological spaces (the underlying spaces of a scheme, Berkovich analytificaiton of schemes, real schemes) are homeomorphic to sets of rational points of a scheme over a hyperfield. Also, recently L. Anderson and J. Davis defined and investigated hyperfield Grassmannians in connection to the MacPhersonian (from oriented matroid theory) in [2].…”

Section: Matroids Over Hyperfieldsmentioning

confidence: 99%

Hopf algebras for matroids over hyperfields

2020

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Recently, M. Baker and N. Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields. Date: December 27, 2017. 2010 Mathematics Subject Classification. 05E99(primary), 16T05(secondary). Key words and phrases. matroid, hyperfield, matroid over hyperfields, Hopf algebra, minor, direct sum.In this sense B and C B carry the same information as C B and B C respectively; these are thus said to determine the same matroid on E "cryptomorphically."Example 2.2. The motivating examples of matroids (hinted at above) are given as follows:(1) Let V be a finite dimensional vector space and E ⊆ V a spanning set of vectors.The bases of V contained in E form the bases of a matroid on E, and the minimal dependent subsets of E form the circuits of a matroid on E. Furthermore, these are the same matroid.(2) Let Γ be a finite, undirected graph with edge set E (loops and parallel edges are allowed). The sets of edges of spanning forests in Γ form the bases of a matroid on E, and the sets of edges of cycles form the circuits of a matroid on E. Furthermore, these are the same matroid (called the graphic matroid of Γ).One can define the notion of isomorphisms of matroids as follows.Definition 2.3. Let M 1 (resp. M 2 ) be a matroid on E 1 (resp. E 2 ) defined by a set B 1 (resp. B 2 ) of bases. We say that M 1 is isomorphic to M 2 if there exists a bijection f :In this case, f is said to be an isomorphism.Example 2.4. Let Γ 1 and Γ 2 be finite graphs and M 1 and M 2 be the corresponding graphic matroids. Every graph isomorphism between Γ 1 and Γ 2 gives rise to a matroid isomorphism between M 1 and M 2 , but the converse need not hold.Recall that given any base B ∈ B(M ) and any element e ∈ E \ B, there is a unique circuit (fundamental circuit ) C B,e of e with respect to B such that C B,e ⊆ B ∪ {e}.One can construct new matroids from given matroids as follows:Definition 2.5 (Direct sum of matroids). Let M 1 and M 2 be matroids on E 1 and E 2 given by bases B 1 and B 2 respectively. The direct sum M 1 ⊕ M 2 is the matroid on E 1 ⊔ E 2 given by the bases B = {B 1 ⊔ B 2 | B i ∈ B i for i = 1, 2}.Remark 2.6. One can easily check that M 1 ⊕ M 2 is indeed a matroid on E 1 ⊔ E 2 .Definition 2.7 (Dual, Restriction, Deletion, and Contraction). Let M be a matroid on a finite set E M with the set B M of bases and the set C M of circuits. Let S be a subset of E M .(1) The dual M * of M is a matroid on E M given by bases

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Geometry of hyperfields

Cited by 9 publications

References 20 publications

Vector bundles on tropical schemes

Vector bundles on tropical schemes

Tropical geometry over the tropical hyperfield

Hopf algebras for matroids over hyperfields

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