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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: A toric surface patch associated with a convex polygon, which has vertices with integer coordinates, is defined, which naturally generalizes classical Bézier surfaces.
Abstract: We define a toric surface patch associated with a convex polygon, which has vertices with integer coordinates. This rational surface patch naturally generalizes classical Bezier surfaces. Several features of toric patches are considered: affine invariance, convex hull property, boundary curves, implicit degree and singular points. The method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed.
92 citations
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TL;DR: In this article, it was shown that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope.
Abstract: A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geometry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.
91 citations
TL;DR: In this article, Kontsevich's homological mirror symmetry conjecture was shown to hold in the context of toric varieties, where the origin point of a toric variety is a convex hull of the primitive vertices of the 1-cones of a simplicial rational polyhedral fan.
Abstract: In this paper we give some evidence for M Kontsevich’s homological mirror symmetry conjecture [13] in the context of toric varieties. Recall that a smooth complete toric variety is given by a simplicial rational polyhedral fan such that jj D R and all maximal cones are non-singular (Fulton [10, Section 2.1]). The convex hull of the primitive vertices of the 1–cones of is a convex polytope which we denote by P , containing the origin as an interior point, and may be thought of as the Newton polytope of a Laurent polynomial W W .C/ ! C. This Laurent polynomial is the Landau–Ginzburg mirror of X .
90 citations
TL;DR: In this article, a functorial normal crossing compactification of the moduli space of smooth cubic surfaces was proposed, analogous to the Grothendieck-Knudsen compactification.
Abstract: We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification \(M_{0,n}\subset\overline{M}_{0,n}\) .
89 citations
TL;DR: In this article, the primitive collections and primitive relations due to Batyrev were used for the classification of nonsingular toric Fano verieties by means of primitive collections.
Abstract: The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano verieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite successions of equivariant blow-ups and blow-downs through nonsingular toric Fano varieties. Especially, we get a new method for the classification of nonsingular toric Fano varieties of dimension at most four. These methods are extended to the case of Gorenstein toric Fano varieties endowed with natural resolutions of singularities. Especially, we easily get a new method for the classification of Gorenstein toric Fano surfaces.
89 citations