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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.
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TL;DR: In this article, it was shown that the natural map from H^0(X,L) tensor H^ 0(M,X,M) to H √ X,L tensor M √ L tensor is surjective, where X is a smooth projective toric surface and L and M are two line bundles on it.
Abstract: Let X be a smooth projective toric surface and L and M two line bundles on X If L is ample and M is generated by global sections, then we show that the natural map from H^0(X,L) tensor H^0(X,M) to H^0(X, L tensor M) is surjective We also consider a generalization to the case when M is an arbitrary line bundle with h^0(X,M) > 0
17 citations
TL;DR: In this paper, the authors consider the metric space of all toric Kahler metrics on a compact toric manifold and obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics.
Abstract: We consider the metric space of all toric Kahler metrics on a compact toric manifold; when “looking at it from infinity” (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding Kahler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.
17 citations
TL;DR: In this article, the authors present arbitrary toric varieties as quotients of quasiaffine toric variety. And they use homogeneous coordinates to express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric.
Abstract: Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas ℚ–Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.
17 citations
TL;DR: In this paper, the existence of homogeneous coordinates for simplicial toric varieties was shown to be equivalent to the Darboux-Jouanolou-Ghys integrability theorem for rational first integrals for one-dimensional foliations.
Abstract: We use the existence of homogeneous coordinates for simplicial toric varieties to prove a result analogous to the Darboux–Jouanolou–Ghys integrability theorem for the existence of rational first integrals for one-dimensional foliations.
17 citations
01 Oct 2013
TL;DR: In this paper, it was shown that Aut(X) acts on X transitively if and only if X is a product of projective spaces, i.e., it can be seen as an automorphism group.
Abstract: Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)-orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces.
17 citations