123 research outputs found
Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models
We show that every rank one smooth Fano threefold has a weak Landau-Ginzburg
model coming from a toric degeneration. The fibers of these Landau-Ginzburg
models can be compactified to K3 surfaces with Picard lattice of rank 19. We
also show that any smooth Fano variety of arbitrary dimension which is a
complete intersection of Cartier divisors in weighted projective space has a
very weak Landau-Ginzburg model coming from a toric degeneration.Comment: v3: minor corrections for final versio
Symplectomorphism group relations and degenerations of Landau-Ginzburg models
In this paper, we describe explicit relations in the symplectomorphism groups
of toric hypersurfaces. To define the elements involved, we construct a proper
stack of toric hypersurfaces with compactifying boundary representing toric
hypersurface degenerations. Our relations arise through the study of the one
dimensional strata of this stack. The results are then examined from the
perspective of homological mirror symmetry where we view sequences of relations
as maximal degenerations of Landau-Ginzburg models. We then study the B-model
mirror to these degenerations, which gives a new mirror symmetry approach to
the minimal model program.Comment: 100 pages, 24 figure
Symmetry in Regular Polyhedra Seen as 2D Möbius Transformations: Geodesic and Panel Domes Arising from 2D Diagrams
This paper shows a methodology for reducing the complex design process of space structures to an adequate selection of points lying on a plane. This procedure can be directly implemented in a bi-dimensional plane when we substitute (i) Euclidean geometry by bi-dimensional projection of the elliptic geometry and (ii) rotations/symmetries on the sphere by Möbius transformations on the plane. These graphs can be obtained by sites, specific points obtained by homological transformations in the inversive plane, following the analogous procedure defined previously in the three-dimensional space. From the sites, it is possible to obtain different partitions of the plane, namely, power diagrams, Voronoi diagrams, or Delaunay triangulations. The first
would generate geo-tangent structures on the sphere; the second, panel structures; and the third, lattice structures
Toric Methods in F-theory Model Building
In this review article we discuss recent constructions of global F-theory GUT
models and explain how to make use of toric geometry to do calculations within
this framework. After introducing the basic properties of global F-theory GUTs
we give a self-contained review of toric geometry and introduce all the tools
that are necessary to construct and analyze global F-theory models. We will
explain how to systematically obtain a large class of compact Calabi-Yau
fourfolds which can support F-theory GUTs by using the software package PALP.Comment: 19 pages. Prepared for the special issue "Computational Algebraic
Geometry in String and Gauge Theory" of Advances in High Energy Physics, v2:
references added, typos correcte
Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau
3-folds starting with (almost) any deformation family of smooth weak Fano
3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau
3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We
pay particular attention to a subclass of weak Fano 3-folds that we call
semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing
theorems and enjoy certain topological properties not satisfied by general weak
Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike
Fanos they often contain P^1s with normal bundle O(-1) + O(-1), giving rise to
compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.
We introduce some general methods to compute the basic topological invariants
of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a
small number of representative examples in detail. Similar methods allow the
computation of the topology in many other examples.
All the features of the ACyl Calabi-Yau 3-folds studied here find application
in arXiv:1207.4470 where we construct many new compact G_2-manifolds using
Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds
constructed from semi-Fano 3-folds are particularly well-adapted for this
purpose.Comment: 107 pages, 1 figure. v3: minor corrections, changed formattin
Enumerative aspects of the Gross-Siebert program
We present enumerative aspects of the Gross-Siebert program in this
introductory survey. After sketching the program's main themes and goals, we
review the basic definitions and results of logarithmic and tropical geometry.
We give examples and a proof for counting algebraic curves via tropical curves.
To illustrate an application of tropical geometry and the Gross-Siebert program
to mirror symmetry, we discuss the mirror symmetry of the projective plane.Comment: A version of these notes will appear as a chapter in an upcoming
Fields Institute volume. 81 page
On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane
For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles in 1970
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