Topic
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 combinatorial proof of White's conjecture that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges is presented by using a lemma proposed by Blasiak.
Abstract: White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.
28 citations
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TL;DR: In this article, it was shown that the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) 6 = 0.
Abstract: In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) 6= 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.
28 citations
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TL;DR: In this paper, the authors studied compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5.
Abstract: We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on $S^2\times S^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.
28 citations
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TL;DR: In this article, the authors proved the conjecture stated in 6, extending and correcting a previous conjecture of Ilardi 5, and classified smooth minimal monomial Togliatti systems of cubics in any dimension.
28 citations
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TL;DR: In this paper, the authors established a correspondence between simplicial fans and certain foliated compact complex manifolds called LVMB-manifolds, and showed that the basic cohomology of the foliation is generated in degree two.
Abstract: We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases.
28 citations