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Displaying similar documents to “Bases for certain varieties of completely regular semigroups”

Relations on a lattice of varieties of completely regular semigroups

Mario Petrich (2020)

Mathematica Bohemica

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Completely regular semigroups 𝒞ℛ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes 𝒞ℛ a variety; its lattice of subvarieties is denoted by ( 𝒞ℛ ) . We study here the relations 𝐊 , T , L and 𝐂 relative to a sublattice Ψ of ( 𝒞ℛ ) constructed in a previous publication. For 𝐑 being any of these relations, we determine the 𝐑 -classes of all varieties in the lattice Ψ as well as the restrictions of 𝐑 to Ψ .

On tangent cones to Schubert varieties in type E

Mikhail V. Ignatyev, Aleksandr A. Shevchenko (2020)

Communications in Mathematics

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We consider tangent cones to Schubert subvarieties of the flag variety G / B , where B is a Borel subgroup of a reductive complex algebraic group G of type E 6 , E 7 or E 8 . We prove that if w 1 and w 2 form a good pair of involutions in the Weyl group W of G then the tangent cones C w 1 and C w 2 to the corresponding Schubert subvarieties of G / B do not coincide as subschemes of the tangent space to G / B at the neutral point.

The Brauer group and the Brauer–Manin set of products of varieties

Alexei N. Skorobogatov, Yuri G. Zahrin (2014)

Journal of the European Mathematical Society

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Let X and Y be smooth and projective varieties over a field k finitely generated over Q , and let X ¯ and Y ¯ be the varieties over an algebraic closure of k obtained from X and Y , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br ( X ¯ ) Br( Y ¯ ) has finite index in the Galois invariant subgroup of Br ( X ¯ × Y ¯ ) . This implies that the cokernel of the natural map Br ( X ) Br ( Y ) Br ( X × Y ) is finite when k is a number field. In this case we prove that the Brauer–Manin set of the...

J -invariant of linear algebraic groups

Viktor Petrov, Nikita Semenov, Kirill Zainoulline (2008)

Annales scientifiques de l'École Normale Supérieure

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Let G be a semisimple linear algebraic group of inner type over a field F , and let X be a projective homogeneous G -variety such that G splits over the function field of X . We introduce the J -invariant of G which characterizes the motivic behavior of X , and generalizes the J -invariant defined by A. Vishik in the context of quadratic forms. We use this J -invariant to provide motivic decompositions of all generically split projective homogeneous G -varieties, e.g. Severi-Brauer varieties,...

Construction of Mendelsohn designs by using quasigroups of ( 2 , q ) -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let q be a positive integer. An algebra is said to have the property ( 2 , q ) if all of its subalgebras generated by two distinct elements have exactly q elements. A variety 𝒱 of algebras is a variety with the property ( 2 , q ) if every member of 𝒱 has the property ( 2 , q ) . Such varieties exist only in the case of q prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property ( 2 , q ) , 2 < q , blocks of Steiner system of type ( 2 , q ) are obtained. The stated correspondence...

On varieties of Hilbert type

Lior Bary-Soroker, Arno Fehm, Sebastian Petersen (2014)

Annales de l’institut Fourier

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A variety X over a field K is of Hilbert type if X ( K ) is not thin. We prove that if f : X S is a dominant morphism of K -varieties and both S and all fibers f - 1 ( s ) , s S ( K ) , are of Hilbert type, then so is X . We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Thélène and Sansuc on algebraic groups.

Elements of large order on varieties over prime finite fields

Mei-Chu Chang, Bryce Kerr, Igor E. Shparlinski, Umberto Zannier (2014)

Journal de Théorie des Nombres de Bordeaux

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Let 𝒱 be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C ( 𝒱 ) such that for almost all primes p for all but at most C ( 𝒱 ) points on the reduction of 𝒱 modulo p at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

Non-supersingular hyperelliptic jacobians

Yuri G. Zarhin (2004)

Bulletin de la Société Mathématique de France

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Let K be a field of odd characteristic p , let f ( x ) be an irreducible separable polynomial of degree n 5 with big Galois group (the symmetric group or the alternating group). Let C be the hyperelliptic curve y 2 = f ( x ) and J ( C ) its jacobian. We prove that J ( C ) does not have nontrivial endomorphisms over an algebraic closure of K if either n 7 or p 3 .

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let k be a field of characteristic p &gt; 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair ( 𝒳 , 𝒜 ) of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group Γ . The space 𝒜 is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism p : 𝒜 𝒳 . The space 𝒜 is equipped with a closed 2 -form, possibly degenerate, and the space 𝒳 has a Poisson structure. The map p is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let  X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in  X . We consider the image 𝒩 1 ( D , X ) of  𝒩 1 ( D ) in  𝒩 1 ( X ) under the natural push-forward of 1 -cycles. We show that ρ X - ρ D codim 𝒩 1 ( D , X ) 8 . Moreover if codim 𝒩 1 ( D , X ) 3 , then either X S × T where S is a Del Pezzo surface, or codim 𝒩 1 ( D , X ) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X - ρ T = 4 .

Geometry of Syzygies via Poncelet Varieties

Giovanna Ilardi, Paola Supino, Jean Vallès (2009)

Bollettino dell'Unione Matematica Italiana

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We consider the Grassmannian 𝔾 r ( k , n ) of ( k + 1 ) -dimensional linear subspaces of V n = H 0 ( 1 , 𝒪 1 ( n ) ) . We define 𝔛 k , r , d as the classifying space of the k -dimensional linear systems of degree n on 1 , whose bases realize a fixed number r of polynomial relations of fixed degree d , say r syzygies of degree d . Firstly, we compute the dimension of 𝔛 k , r , d . In the second part we make a link between 𝔛 k , r , d and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...

On the real X -ranks of points of n ( ) with respect to a real variety X n

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let  X n be an integral and non-degenerate m -dimensional variety defined over . For any P n ( ) the real X -rank r X , ( P ) is the minimal cardinality of S X ( ) such that P S . Here we extend to the real case an upper bound for the X -rank due to Landsberg and Teitler.

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

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We take up the study of the Brill-Noether loci W r ( L , X ) : = { η Pic 0 ( X ) | h 0 ( L η ) r + 1 } , where X is a smooth projective variety of dimension > 1 , L Pic ( X ) , and r 0 is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for h 0 ( K D ) , where D is a divisor that moves linearly on a smooth projective variety X of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension > 2 . In the 2 -dimensional case...

Involutivity degree of a distribution at superdensity points of its tangencies

Silvano Delladio (2021)

Archivum Mathematicum

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Let Φ 1 , ... , Φ k + 1 (with k 1 ) be vector fields of class C k in an open set U N + m , let 𝕄 be a N -dimensional C k submanifold of U and define 𝕋 : = { z 𝕄 : Φ 1 ( z ) , ... , Φ k + 1 ( z ) T z 𝕄 } where T z 𝕄 is the tangent space to 𝕄 at z . Then we expect the following property, which is obvious in the special case when z 0 is an interior point (relative to 𝕄 ) of 𝕋 : If z 0 𝕄 is a ( N + k ) -density point (relative to 𝕄 ) of 𝕋 then all the iterated Lie brackets of order less or equal to k Φ i 1 ( z 0 ) , [ Φ i 1 , Φ i 2 ] ( z 0 ) , [ [ Φ i 1 , Φ i 2 ] , Φ i 3 ] ( z 0 ) , ... ( h , i h k + 1 ) belong to T z 0 𝕄 . Such a property has been proved in [9] for k = 1 and its proof in the...