Tropical intersection products on smooth varieties

Lars Allermann

Displaying similar documents to “Tropical intersection products on smooth varieties”

Toric and tropical compactifications of hyperplane complements

Graham Denham (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel $^{'}$ fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.

Minimal and minimum size latin bitrades of each genus

James Lefevre, Diane Donovan, Nicholas J. Cavenagh, Aleš Drápal (2007)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that $T^{\circ}$ and $T^{☆}$ are partial latin squares of order $n$ , with the property that each row and each column of $T^{\circ}$ contains the same set of entries as the corresponding row or column of $T^{☆}$ . In addition, suppose that each cell in $T^{\circ}$ contains an entry if and only if the corresponding cell in $T^{☆}$ contains an entry, and these entries (if they exist) are different. Then the pair $T = (T^{\circ}, T^{☆})$ forms a . The of $T$ is the total number of filled cells in $T^{\circ}$ (equivalently $T^{☆}$ ). The latin bitrade is if there is no...

Asymptotics of eigensections on toric varieties

A. Huckleberry, H. Sebert (2013)

Annales de l’institut Fourier

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Using exhaustion properties of invariant plurisubharmonic functions along with basic combinatorial information on toric varieties, we prove convergence results for sequences of densities $| ϕ_{n} |^{2} = | s_{N} |^{2} / | | s_{N} {| |}_{L^{2}}^{2}$ for eigensections $s_{N} \in Γ (X, L^{N})$ approaching a semiclassical ray. Here $X$ is a normal compact toric variety and $L$ is an ample line bundle equipped with an arbitrary positive bundle metric which is invariant with respect to the compact form of the torus. Our work was motivated by and extends that of Shiffman, Tate...

A note on arc-disjoint cycles in tournaments

Jan Florek (2014)

Colloquium Mathematicae

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We prove that every vertex v of a tournament T belongs to at least $m a x m i n δ ⁺ (T), 2 δ ⁺ (T) - d ⁺_{T} (v) + 1, m i n δ ¯ (T), 2 δ ¯ (T) - d ¯_{T} (v) + 1$ arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and $d ⁺_{T} (v)$ (or $d ¯_{T} (v)$ ) is the out-degree (resp. in-degree) of v.

Arithmetic of 0-cycles on varieties defined over number fields

Yongqi Liang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a rationally connected algebraic variety, defined over a number field $k .$ We find a relation between the arithmetic of rational points on $X$ and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for $K$ -rational points on $X_{K}$ for all finite extensions $K / k;$ (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles...

Cycle-pancyclism in bipartite tournaments I

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ [(n+4)/2], there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 3$ and the result is best possible. In a forthcoming paper the case of directed cycles of length k, k even and k <...

Cycle-pancyclism in bipartite tournaments II

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T, and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T[V(γ)]? It is proved that for an even k in the range (n+6)/2 ≤ k ≤ n-2, there exists a directed cycle $C_{h (k)}$ of length h(k), h(k) ∈ k,k-2 with $| A (C_{h (k)}) \cap A (γ) | \geq h (k) - 4$ and the result is best possible. In a previous paper a similar result for 4 ≤ k ≤ (n+4)/2 was...

Finiteness of cominuscule quantum $K$ -theory

Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, Nicolas Perrin (2013)

Annales scientifiques de l'École Normale Supérieure

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The product of two Schubert classes in the quantum $K$ -theory ring of a homogeneous space $X = G / P$ is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on $X$ . We show that if $X$ is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to $X$ that take the marked points to general Schubert varieties and...

Pairs of forbidden class of subgraphs concerning $K_{1, 3}$ and P₆ to have a cycle containing specified vertices

Takeshi Sugiyama, Masao Tsugaki (2009)

Discussiones Mathematicae Graph Theory

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In [3], Faudree and Gould showed that if a 2-connected graph contains no $K_{1, 3}$ and P₆ as an induced subgraph, then the graph is hamiltonian. In this paper, we consider the extension of this result to cycles passing through specified vertices. We define the families of graphs which are extension of the forbidden pair $K_{1, 3}$ and P₆, and prove that the forbidden families implies the existence of cycles passing through specified vertices.

Categorical equivalences of varieties generated by algebras $𝔄$ and $𝔄^{r}$

K. Denecke (1988)

Banach Center Publications

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On the differential geometry of some classes of infinite dimensional manifolds

Maysam Maysami Sadr, Danial Bouzarjomehri Amnieh (2024)

Archivum Mathematicum

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Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $Γ_{X}$ of any manifold $X$ . The name comes from the fact that various elements of the geometry of $Γ_{X}$ are constructed via lifting of the corresponding elements of the geometry of $X$ . In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$ . In order to define...

The finiteness of compact varieties in $C^{n}$

Walter Rudin (1981)

Annales Polonici Mathematici

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A note on normal varieties of monounary algebras

Ivan Chajda, Helmut Länger (2002)

Czechoslovak Mathematical Journal

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A variety is called normal if no laws of the form $s = t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $(A, f)$ and let $V$ be a non-trivial non-normal element of $L$ . Then $V$ is of the form $M o d (f^{n} (x) = x)$ with some $n > 0$ . It is shown that the smallest normal variety containing $V$ is contained in $H S C (M o d (f^{m n} (x) = x))$ for every $m > 1$ where $C$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal...

Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, $W \subset k^{n + 1}$ , and $F : k ⁿ \to k^{n + 1}$ such that ${F |}_{V} = f$ and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, $W \subset k^{m}$ with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

Interpolating discrete multiplicity varieties for $A ⁰_{p} (ℂ ⁿ)$

Bao Qin Li, Enrique Villamor (2006)

Studia Mathematica

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A necessary and sufficient condition is obtained for a discrete multiplicity variety to be an interpolating variety for the space $A ⁰_{p} (ℂ ⁿ)$ .