Finiteness of cominuscule quantum $K$-theory

Anders S. Buch; Pierre-Emmanuel Chaput; Leonardo C. Mihalcea; Nicolas Perrin

Displaying similar documents to “Finiteness of cominuscule quantum $K$ -theory”

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...

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 $f : X \to S$ is a dominant morphism of $K$ -varieties and both $S$ and all fibers $f^{- 1} (s)$ , $s \in S (K)$ , 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.

Bases for certain varieties of completely regular semigroups

Mario Petrich (2021)

Commentationes Mathematicae Universitatis Carolinae

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Completely regular semigroups equipped with the unary operation of inversion within their maximal subgroups form a variety, denoted by $𝒞ℛ$ . The lattice of subvarieties of $𝒞ℛ$ is denoted by $ℒ (𝒞ℛ)$ . For each variety in an $⋂$ -subsemilattice $Γ$ of $ℒ (𝒞ℛ)$ , we construct at least one basis of identities, and for some important varieties, several. We single out certain remarkable types of bases of general interest. As an application for the local relation $L$ , we construct $𝐋$ -classes of all varieties in $Γ$ . Two...

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 $\bar{X}$ and $\bar{Y}$ 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 $(\bar{X}) \oplus$ Br( $\bar{Y})$ has finite index in the Galois invariant subgroup of Br $(\bar{X} \times \bar{Y})$ . This implies that the cokernel of the natural map Br $(X) \oplus$ Br $(Y) \to$ Br $(X \times Y)$ 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,...

Polarizations of Prym varieties for Weyl groups via abelianization

Herbert Lange, Christian Pauly (2009)

Journal of the European Mathematical Society

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Let $π : Z \to X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group $G$ . For any dominant weight $λ$ consider the curve $Y = Z / Stab (λ)$ . The Kanev correspondence defines an abelian subvariety $P_{λ}$ of the Jacobian of $Y$ . We compute the type of the polarization of the restriction of the canonical principal polarization of $Jac (Y)$ to $P_{λ}$ in some cases. In particular, in the case of the group $E_{8}$ we obtain families of Prym-Tyurin varieties. The main idea is...

On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

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In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $- K_{S} . C + p_{g} (C) - 1$ , where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $𝚍𝚒𝚖 (V) = - K_{S} . C + p_{g} (C) - 1$ does not imply the nodality of $C$ even if $C$ belongs...

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 $\mathbb{G}r(k,n)$ of $(k+1)$ -dimensional linear subspaces of $V_{n}=H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}_{1}}(n))$ . We define $\mathfrak{X}_{k,r,d}$ as the classifying space of the $k$ -dimensional linear systems of degree $n$ on $\mathbb{P}^{1}$ , 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 $\mathfrak{X}_{k,r,d}$ . In the second part we make a link between $\mathfrak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

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Consider the families of curves $C^{n, A} : y ² = x ⁿ + A x$ and $C_{n, A} : y ² = x ⁿ + A$ where A is a nonzero rational. Let $J^{n, A}$ and $J_{n, A}$ denote their respective Jacobian varieties. The torsion points of $C^{3, A} (ℚ)$ and $C_{3, A} (ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7, A} (ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7, A} (ℚ) [2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3, A}$ (A ≠ 4) and $J^{5, A}$ . We...

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.

Singularities of theta divisors and the geometry of $𝒜_{5}$

Gavril Farkas, Samuele Grushevsky, Salvati R. Manni, Alessandro Verra (2014)

Journal of the European Mathematical Society

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We study the codimension two locus $H$ in $𝒜_{g}$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $[H] \in C H^{2} (𝒜_{g})$ for every $g$ . For $g = 4$ , this turns out to be the locus of Jacobians with a vanishing theta-null. For $g = 5$ , via the Prym map we show that $H \subset 𝒜_{5}$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\bar{𝒜_{5}}$ and show that the component $\bar{N_{0}^{'}}$ of the Andreotti-Mayer...

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...

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 $n \geq 5$ 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 $n \geq 7$ or $p \neq 3$ .

Quiver varieties and the character ring of general linear groups over finite fields

Emmanuel Letellier (2013)

Journal of the European Mathematical Society

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Given a tuple $(𝒳_{1}, ..., 𝒳_{k})$ of irreducible characters of $G L_{n} (F_{q})$ we define a star-shaped quiver $Γ$ together with a dimension vector $v$ . Assume that $(𝒳_{1}, ..., 𝒳_{k})$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product $𝒳_{1} \otimes \dots \otimes 𝒳_{k}$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(Γ, v)$ . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied,...

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 $p : 𝒜 \to 𝒳$ . 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...

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.

Unique decomposition for a polynomial of low rank

Edoardo Ballico, Alessandra Bernardi (2013)

Annales Polonici Mathematici

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Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of $ℙ^{m}$ into $ℙ^{\binom{m + d}{d} - 1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F = M ₁^{d} + \dots + M_{t}^{d} + Q$ , where $M ₁, . . ., M_{t}$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q = \sum_{i = 1}^{q} l_{i}^{d - d_{i}} m_{i}$ with $l_{i}$ ’s linear forms...

Generating varieties for the triple loop space of classical Lie groups

Yasuhiko Kamiyama (2003)

Fundamenta Mathematicae

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For G = SU(n), Sp(n) or Spin(n), let $C_{G} (S U (2))$ be the centralizer of a certain SU(2) in G. We have a natural map $J : G / C_{G} (S U (2)) \to Ω ₀ ³ G$ . For a generator α of $H ⁎ (G / C_{G} (S U (2)); ℤ / 2)$ , we describe J⁎(α). In particular, it is proved that $J ⁎ : H ⁎ (G / C_{G} (S U (2)); ℤ / 2) \to H ⁎ (Ω ₀ ³ G; ℤ / 2)$ is injective.

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a uniruled projective manifold and let $x$ be a general point. The main result of [2] says that if the $(- K_{X})$ -degrees (i.e., the degrees with respect to the anti-canonical bundle of $X$ ) of all rational curves through $x$ are at least $dim X + 1$ , then $X$ is a projective space. In this paper, we study the structure of $X$ when the $(- K_{X})$ -degrees of all rational curves through $x$ are at least $dim X$ . Our study uses the projective variety $𝒞_{x} \subset ℙ T_{x} (X)$ , called the VMRT at $x$ , defined as the union of tangent directions to the...

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...

Displaying similar documents to “Finiteness of cominuscule quantum K -theory”

Displaying similar documents to “Finiteness of cominuscule quantum $K$ -theory”