Constructions and complexity of secondary polytopes

Billera, Louis J.; Filliman, P.; Sturmfels, Bernd

doi:10.1016/0001-8708(90)90077-z

Cited by 149 publications

(195 citation statements)

References 12 publications

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“…the generators of the Mori cone can be constructed as described in [31], [16] As the next step we set up the∆ * -hypergeometric system (see e.g. [26][16] for details).…”

Section: Selected Examplesmentioning

confidence: 99%

“…For instance, we cannot just invert z 5 (which would correspond to the replacement l (5) A → −l (5) A ) without generating inhomogeneous terms in I(θ z ), which is incompatible with the required ring structure of R. It is easy to see that the only possibilities are to invert z 2 , z 3 , z 4 and z 4 z 5 , accompanied by transformations of the other variables. These transformations correspond to the flops leading to the coordinate patches described by the Mori cone of subdivisions B, C, D and E. They form part of the secondary fan as described in [31].…”

Section: (525)mentioning

confidence: 99%

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Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

et al. 1995

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Abstract:We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton-corrected Yukawa couplings, and the topological one-loop partition function to the case of complete intersections with higher-dimensional moduli spaces. We will develop a new method of obtaining the instanton-corrected Yukawa couplings through a study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the Kähler moduli fields induced from the ambient space for all complete intersections in non singular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models that are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces.

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“…the generators of the Mori cone can be constructed as described in [31], [16] As the next step we set up the∆ * -hypergeometric system (see e.g. [26][16] for details).…”

Section: Selected Examplesmentioning

confidence: 99%

Section: (525)mentioning

confidence: 99%

Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

et al. 1995

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“…If not, every triangulation that refines T w or T −w is, respectively, minimal or maximal. 11 The same is true for non-regular triangulations. The point v T is well-defined, via the Gel'fand-KapranovZelevinskii coordinates for the secondary polytope, even if T is not regular.…”

Section: The Flippable Circuit Is ({A} τ )mentioning

confidence: 83%

“…Then, it is no surprise that for the regular triangulations T 1 and T 2 corresponding to vertices v T 1 and v T 2 of the secondary polytope one has 11 : Figure 2), except perturbed by slightly rotating the interior triangle "123" counter-clockwise. That is,…”

Section: The Flippable Circuit Is ({A} τ )mentioning

confidence: 99%

Geometric bistellar ﬂips: the setting, the context and a construction

Santos¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract.We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas.As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected. Mathematics Subject Classification (2000). Primary 52B11; Secondary 52B20.

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“…We therefore expect simple relations of the generators of the Mori cones and large complex structure coordinates for all subdivisions to those of subdivision A. After extending the vectors vi* to ~i* = (1, vi*), the generators of the Mori cone can be constructed as described in [30,16] They define the large complex structure variables as z~ A) = -a l~) and z~ A) = a l~' for i = 2, 3, 4, 5. As a next step, we set up the A*-hypergeometric system (see e.g.…”

Section: Selected Examplesmentioning

confidence: 99%

[Experimental nuclear physics]

1992

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We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton corrected Yukawa couplings and the topological one-loop partition function to the case of complete intersections with higher dimensional moduli spaces. We will develop a new method of obtaining the instanton corrected Yukawa couplings through a study of the solutions of the Picard-Fuchs equations. This leads to closed formulas for the prepotential for the K~ihler moduli fields induced from the ambient space for all complete intersections in nonsingular weighted projective spaces. As examples we treat part of the moduli space of the phenomenologically interesting three-generation models which are found in this class. We also apply our method to solve the simplest model in which a topology change was observed and discuss examples of complete intersections in singular ambient spaces.

show abstract

Constructions and complexity of secondary polytopes

Cited by 149 publications

References 12 publications

Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces

Geometric bistellar ﬂips: the setting, the context and a construction

[Experimental nuclear physics]

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