On the conjecture of King for smooth toric Deligne–Mumford stacks

Borisov, Lev A.; Hua, Zheng

doi:10.1016/j.aim.2008.11.017

Cited by 42 publications

(62 citation statements)

References 11 publications

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“…In this particularly simple case, the desired quasi-equivalence can be deduced from the computation in Section 3.4 as all line bundles on P 2 (1, 1, m) are isomorphic to O(kD (−1,0) ) for some k ∈ Z and have sections matching with integer points of the corresponding polytope. Alternatively, the methods of Section 3.5 also apply in this setting as the moment polytope of P 2 (1, 1, m) is simplicial and the cohomology of line bundles on a smooth toric Deligne-Mumford stack can be described analogously to the case of smooth toric varieties as shown in Proposition 4.1 of [10]. As before, we expect that we should also have a quasi-equivalence between T w π F ∆ t m (W Σm ) and a dg-enhancement of D b Coh(P 2 (1, 1, m)) when F ∆ t m (W Σm ) has enough objects.…”

Section: Tropical Divisions Versus Combinatorial Divisionsmentioning

confidence: 99%

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Hanlon

2019

Advances in Mathematics

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Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category of a Laurent polynomial on (C * ) n , which has other potential applications, and give evidence of homological mirror symmetry for non-compact toric varieties.

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Section: Tropical Divisions Versus Combinatorial Divisionsmentioning

confidence: 99%

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Hanlon

2019

Advances in Mathematics

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“…In Sections 3.2.2 and 8.3 of [EF], Eager and Franco discuss a possible coordinate system for working with mutation sequences of quivers from brane tilings. They are motivated by previous work in tilting theory [BH09] and the multi-dimensional octahedron recurrence [HK06,HS10,S07], and sketch examples of coordinates for dP 2 and dP 3 . In their coordinate system, they describe certain duality cascades that act as translations of a zonotope.…”

Section: 5mentioning

confidence: 99%

Beyond Aztec Castles: Toric Cascades in the dP 3 Quiver

Lai

Musiker

2017

Commun. Math. Phys.

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Abstract. Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi-Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface dP 3 . In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables which are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the dP 3 brane tiling for these formulas in most cases.

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“…It was disproved [6,7,11] in infinitely many cases. They proved Conjecture 1.1 [3] in the case of Fano stacks for which either Picard number or dimension is at most two. Borisov and Hua have proposed the following modification (and a generalization).…”

Section: Introductionmentioning

confidence: 99%

“…Borisov and Hua have proposed the following modification (and a generalization). They proved Conjecture 1.1 [3] in the case of Fano stacks for which either Picard number or dimension is at most two. The case of nef-Fano Del Pezzo stacks was further treated in [8].…”

Section: Introductionmentioning

confidence: 99%

Maximal lengths of exceptional collections of line bundles

Efimov

2014

Journal of the London Mathematical Society

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In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant c>34 there exist infinitely many toric Fano varieties Y with Picard number three, such that the maximal length of exceptional collection of line bundles on Y is strictly less than crkK0(Y). To obtain varieties without full exceptional collections of line bundles, it suffices to put c=1. On the other hand, we prove that for any toric nef‐Fano DM stack Y with Picard number three, there exists a strong exceptional collection of line bundles on Y of length at least 34rkK0(Y). The constant 34 is thus maximal with this property.

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On the conjecture of King for smooth toric Deligne–Mumford stacks

Cited by 42 publications

References 11 publications

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

Beyond Aztec Castles: Toric Cascades in the dP 3 Quiver

Maximal lengths of exceptional collections of line bundles

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