On the dimension of secant varieties

Luca Chiantini; Ciro Ciliberto

Displaying similar documents to “On the dimension of secant varieties”

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

Walter Rudin (1981)

Annales Polonici Mathematici

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The dimension of a variety

Ewa Graczyńska, Dietmar Schweigert (2007)

Discussiones Mathematicae - General Algebra and Applications

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Derived varieties were invented by P. Cohn in [4]. Derived varieties of a given type were invented by the authors in [10]. In the paper we deal with the derived variety $V_{σ}$ of a given variety, by a fixed hypersubstitution σ. We introduce the notion of the dimension of a variety as the cardinality κ of the set of all proper derived varieties of V included in V. We examine dimensions of some varieties in the lattice of all varieties of a given type τ. Dimensions of varieties of lattices...

Equivariant K-theory of flag varieties revisited and related results

V. Uma (2013)

Colloquium Mathematicae

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We obtain several several results on the multiplicative structure constants of the T-equivariant Grothendieck ring $K_{T} (G / B)$ of the flag variety G/B. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T} (G / B)$ to R(T) ⊗ R(T), where R(T) denotes the representation ring of the torus T. We further apply our results to describe the multiplicative structure constants of $K {(X)}_{ℚ}$ where X denotes the wonderful compactification of the adjoint group of G, in terms of the structure...

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Н.Я. Медведев (1983)

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Some topological conditions for projective algebraic manifolds with degenerate dual varieties: connections with $\mathbf{P}$ -bundles

Antonio Lanteri, Daniele Struppa (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si illustrano alcune relazioni tra le varietà proiettive complesse con duale degenere, le varietà la cui topologia si riflette in quella della sezione iperpiana in misura maggiore dell'ordinario e le varietà fibrate in spazi lineari su di una curva.

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^{n} A$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^{n} A$ carries a divisor with vanishing Beauville-Bogomolov square.

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...

Derived invariance of the number of holomorphic $1$ -forms and vector fields

Mihnea Popa, Christian Schnell (2011)

Annales scientifiques de l'École Normale Supérieure

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We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.

Varieties and finite closure conditions

John T. Baldwin, Joel Berman (1976)

Colloquium Mathematicae

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

K. Denecke (1988)

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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...

Complexity of hypersubstitutions and lattices of varieties

Thawhat Changphas, Klaus Denecke (2003)

Discussiones Mathematicae - General Algebra and Applications

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Hypersubstitutions are mappings which map operation symbols to terms. The set of all hypersubstitutions of a given type forms a monoid with respect to the composition of operations. Together with a second binary operation, to be written as addition, the set of all hypersubstitutions of a given type forms a left-seminearring. Monoids and left-seminearrings of hypersubstitutions can be used to describe complete sublattices of the lattice of all varieties of algebras of a given type. The...

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 $A ⁰_{p} (ℂ ⁿ)$ .

Schubert varieties in $G / B \times G / B$

V. B. Mehta, A. Ramanathan (1988)

Compositio Mathematica

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k-Normalization and (k+1)-level inflation of varieties

Valerie Cheng, Shelly Wismath (2008)

Discussiones Mathematicae - General Algebra and Applications

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Let τ be a type of algebras. A common measurement of the complexity of terms of type τ is the depth of a term. For k ≥ 1, an identity s ≈ t of type τ is said to be k-normal (with respect to this depth complexity measurement) if either s = t or both s and t have depth ≥ k. A variety is called k-normal if all its identities are k-normal. Taking k = 1 with respect to the usual depth valuation of terms gives the well-known property of normality of identities or varieties. For any variety...

$G$ -тождества и $G$ -многообразия

М.Г. Амаглобели, В.Н. Ремесленников (2000)

Algebra i Logika

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Continuity of the relative extremal function on analytic varieties in ℂⁿ

Frank Wikström (2005)

Annales Polonici Mathematici

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Let V be an analytic variety in a domain Ω ⊂ ℂⁿ and let K ⊂ ⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V, we prove that the relative extremal function $ω_{K}$ is continuous on V if Ω is hyperconvex and K is regular.

Smooth double subvarieties on singular varieties, III

M. R. Gonzalez-Dorrego (2016)

Banach Center Publications

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Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that rC = S ∩ F, r ∈ ℕ, where S and F are two surfaces and all the singularities of F are of the form $z ³ = x^{3 s} - y^{3 s}$ , s ∈ ℕ. We prove that C can never pass through such kind of singularities of a surface, unless r = 3a, a ∈ ℕ. We study multiplicity-r structures on varieties r ∈ ℕ. Let Z be a reduced irreducible nonsingular (n-1)-dimensional variety such that rZ = X ∩ F, where X is a normal n-fold, F...

A new necessary condition for analytic varieties satisfying the local Phragmén-Lindelöf condition

Tobias Heinrich (2005)

Annales Polonici Mathematici

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For an analytic variety V in ℂⁿ containing the origin which satisfies the local Phragmén-Lindelöf condition $P L_{l o c} (0)$ it is shown that for each real simple curve γ and each d ≥ 1 the limit variety $T_{γ, d} V$ satisfies the strong Phragmén-Lindelöf condition (SPL).

Algebraic cobordism of bundles on varieties

Y.-P. Lee, Rahul Pandharipande (2012)

Journal of the European Mathematical Society

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The double point relation defines a natural theory of algebraic cobordism for bundles on varieties. We construct a simple basis (over $ℚ$ ) of the corresponding cobordism groups over Spec( $ℂ$ ) for all dimensions of varieties and ranks of bundles. The basis consists of split bundles over products of projective spaces. Moreover, we prove the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism.