770 research outputs found
An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts
Even a cursory inspection of the Hodge plot associated with Calabi-Yau
threefolds that are hypersurfaces in toric varieties reveals striking
structures. These patterns correspond to webs of elliptic-K3 fibrations whose
mirror images are also elliptic-K3 fibrations. Such manifolds arise from
reflexive polytopes that can be cut into two parts along slices corresponding
to the K3 fibers. Any two half-polytopes over a given slice can be combined
into a reflexive polytope. This fact, together with a remarkable relation on
the additivity of Hodge numbers, explains much of the structure of the observed
patterns.Comment: 30 pages, 15 colour figure
Gorenstein toric Fano varieties
We investigate Gorenstein toric Fano varieties by combinatorial methods using
the notion of a reflexive polytope which appeared in connection to mirror
symmetry. The paper contains generalisations of tools and previously known
results for nonsingular toric Fano varieties. As applications we obtain new
classification results, bounds of invariants and formulate conjectures
concerning combinatorial and geometrical properties of reflexive polytopes.Comment: AMS-LaTeX, 29 pages with 5 figure
Lattice polytopes of degree 2
A theorem of Scott gives an upper bound for the normalized volume of lattice
polygons with exactly interior lattice points. We will show that the same
bound is true for the normalized volume of lattice polytopes of degree 2 even
in higher dimensions. In particular, there is only a finite number of quadratic
polynomials with fixed leading coefficient being the -polynomial of a
lattice polytope.Comment: 8 page
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