6,227 research outputs found
From Cracked Polytopes to Fano Threefolds
We construct Fano threefolds with very ample anti-canonical bundle and Picard
rank greater than one from cracked polytopes - polytopes whose intersection
with a complete fan forms a set of unimodular polytopes - using Laurent
inversion; a method developed jointly with Coates-Kasprzyk. We also give
constructions of rank one Fano threefolds from cracked polytopes, following
work of Christophersen-Ilten and Galkin. We explore the problem of classifying
polytopes cracked along a given fan in three dimensions, and classify the
unimodular polytopes which can occur as 'pieces' of a cracked polytope.Comment: New introduction and section on the connection with the Gross-Siebert
program. 46 page
Discrete convexity and unimodularity. I
In this paper we develop a theory of convexity for a free Abelian group M
(the lattice of integer points), which we call theory of discrete convexity. We
characterize those subsets X of the group M that could be call "convex". One
property seems indisputable: X should coincide with the set of all integer
points of its convex hull co(X) (in the ambient vector space V). However, this
is a first approximation to a proper discrete convexity, because such
non-intersecting sets need not be separated by a hyperplane. This issue is
closely related to the question when the intersection of two integer polyhedra
is an integer polyhedron. We show that unimodular systems (or more generally,
pure systems) are in one-to-one correspondence with the classes of discrete
convexity. For example, the well-known class of g-polymatroids corresponds to
the class of discrete convexity associated to the unimodular system A_n:={\pm
e_i, e_i-ej} in Z^n.Comment: 26 pages, Late
Intersection Theory on Linear Subvarieties of Toric Varieties
We give a complete description of the cohomology ring of a
compactification of a linear subvariety of a torus in a smooth toric
variety whose fan is supported on the tropicalization of . It turns
out that cocycles on canonically correspond to Minkowski weights
on and that the cup product is described by the intersection product
on the tropical matroid variety .Comment: published versio
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