Cocycles on tropical varieties via piecewise polynomials

Francois, Georges

doi:10.1090/s0002-9939-2012-11359-0

Cited by 15 publications

(19 citation statements)

References 9 publications

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“…In this section, we generalize the notion of a tropical cycle allowing smooth real functions on the maximal faces as weights. This is similar to the tropical fans with polynomial weights introduced by Esterov and François [Est12,Fra13]. We generalize basic facts from stable tropical intersection theory and introduce the corner locus of a piecewise smooth function.…”

Section: Tropical Intersection Theory With Smooth Weightsmentioning

confidence: 84%

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A tropical approach to nonarchimedean Arakelov geometry

Gubler

Kuennemann

2017

Alg. Number Th.

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Abstract. Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called δ-forms on the non-archimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincaré-Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a δ-form. We introduce the associated Monge-Ampère measure µ as a wedge-power of this first Chern δ-form and we show that µ is equal to the corresponding Chambert-Loir measure. The * -product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of δ-forms, we obtain a non-archimedean analogue at least in the case of divisors. We use it to compute non-archimedean local heights of proper varieties.

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Section: Tropical Intersection Theory With Smooth Weightsmentioning

confidence: 84%

“…Similarly as in [Est12,Fra13], we introduce the corner locus of a piecewise smooth function. Definition 1.10 (Corner locus).…”

Section: Tropical Intersection Theory With Smooth Weightsmentioning

confidence: 99%

A tropical approach to nonarchimedean Arakelov geometry

Gubler

Kuennemann

2017

Alg. Number Th.

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“…Example 3. 12 We now want to see that this is the strongest possible statement (see also the subsequent polymake example). x, p) is not locally irreducible: it contains the two lines…”

Section: Irreducibilitymentioning

confidence: 93%

“…We choose parameters x = (1, 1, 1, 1, −4) and 4} . In this case, Lemma 3.19 tells us that the value of the function at C is |a 2 − a 1 | + |a 3 − a 1 | + |a 3 − a 2 | = 1 + 2 + 1 = 4 Now, we define the following divisor of a piecewise polynomial (see, for example, [12] for a treatment of piecewise polynomials. For now it suffices if we define them as sums of products of rational functions):…”

Section: Definition 321mentioning

confidence: 99%

“…3.1, we study whether tropical Hurwitz cycles are connected in codimension one. We give a combinatorial proof of the following result: [12] that each subcycle of a matroidal fan (such as M trop 0,n ) can be written as the sum of products of rational functions, but the result is non-constructive. We show that H trop n−4 (x, 0) can be cut out by the rational function that adds up distances of vertex images of covers.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorics of tropical Hurwitz cycles

Hampe¹

2015

J Algebr Comb

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We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a sum of boundary divisors. The results and counterexamples in this paper were obtained with the help of a-tint, an extension for polymake for tropical intersection theory.Comment: 29 pages, 16 figures. Minor revisions, to appear in Journal of Algebraic Combinatoric

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Moduli Spaces of Curves in Tropical Varieties

Gathmann

Ochse

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental classes in algebraic geometry. As we focus on the combinatorial aspect, we take the weights on certain basic 0-dimensional local combinatorial curve types as input data, and give a compatibility condition in dimension 1 to ensure that this input data glues to a global well-defined tropical cycle. As an application, we construct such moduli spaces for the case of lines in surfaces, and in a subsequent paper for stable maps to a curve [GMO17].

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Cocycles on tropical varieties via piecewise polynomials

Abstract: ABSTRACT. We use piecewise polynomials to define tropical cocycles generalising the well-known notion of Cartier divisors to higher codimensions. We also introduce an intersection product of cocycles with tropical cycles and prove that this gives rise to a Poincaré duality in some cases.

Cited by 15 publications

References 9 publications

A tropical approach to nonarchimedean Arakelov geometry

A tropical approach to nonarchimedean Arakelov geometry

Combinatorics of tropical Hurwitz cycles

Moduli Spaces of Curves in Tropical Varieties

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