Toric variety

About: Toric variety is a research topic. Over the lifetime, 2630 publications have been published within this topic receiving 65604 citations. The topic is also known as: torus embedding.

A basis for the symplectic group branching algebra

Abstract: The symplectic group branching algebra, $\mathcal {B}$ , is a graded algebra whose components encode the multiplicities of irreducible representations of Sp2n?2(?) in each finite-dimensional irreducible representation of Sp2n (?). By describing on $\mathcal {B}$ an ASL structure, we construct an explicit standard monomial basis of $\mathcal {B}$ consisting of Sp2n?2(?) highest weight vectors. Moreover, $\mathcal {B}$ is known to carry a canonical action of the n-fold product SL2×?×SL2, and we show that the standard monomial basis is the unique (up to scalar) weight basis associated to this representation. Finally, using the theory of Hibi algebras we describe a deformation of $\mathrm{Spec}(\mathcal {B})$ into an explicitly described toric variety.

Equivariant class group. III. Almost principal fibrer bundles

Abstract: As a formulation of 'codimension-two arguments' in invariant theory, we define a (rational) almost principal bundle. It is a principal bundle off closed subsets of codimension two or more. We discuss the behavior of the category of reflexive modules over locally Krull schemes, the category of the coherent sheaves which satisfy Serre's condition $(S'_2)$ over Noetherian $(S_2)$ schemes with dualizing complexes, the class group, the canonical module, the Frobenius pushforwards, and global $F$-regularity, of a rational almost principal bundle. We give examples of finite group schemes, multisection rings, surjectively graded rings, and determinantal rings, and give unified treatment and new proofs to known results in invariant theory, algebraic geometry, and commutative algebra, and generalize some of them. In particular, we generalize the result on the canonical module of the multisection ring of a sequence of divisors by Kurano and the author. We also give a new proof of a generalization of Thomsen's result on the Frobenius pushforwards of the structure sheaf of a toric variety.

Rational cohomology of the real Coxeter toric variety of type A

Abstract: The toric variety corresponding to the Coxeter fan of type A can also be described as a De Concini-Procesi wonderful model. Using a general result of Rains which relates cohomology of real De Concini-Procesi models to poset homology, we give formulas for the Betti numbers of the real toric variety, and the symmetric group representations on the rational cohomologies. We also show that the rational cohomology ring is not generated in degree 1.

Seidel elements and mirror transformations

Abstract: The goal of this article is to give a precise relation between the mirror symmetry transformation of Givental and the Seidel elements for a smooth projective toric variety X with −KX nef. We show that the Seidel elements entirely determine the mirror transformation and mirror coordinates.

Fonctions ZÊta Des Hauteurs Des Espaces Fibrés

Abstract: In this paper we study the compatibility of Manin’s conjectures concerning asymptotics of rational points on algebraic varieties with certain natural geometric constructions. More precisely, we consider locally trivial fibrations constructed from torsors under linear algebraic groups. The main problem is to understand the behaviour of the height function as one passes from fiber to fiber - a difficult problem, even though all fibers are isomorphic. We will be mostly interested in fibrations induced from torsors under split tori. Asymptotic properties follow from analytic properties of height zeta functions. Under reasonable assumptions on the analytic behaviour of the height zeta function for the base we establish analytic properties of the height zeta function of the total space.