3,205 research outputs found
Toric Fano 3-folds with terminal singularities
This paper classifies all toric Fano 3-folds with terminal singularities.
This is achieved by solving the equivalent combinatoric problem; that of
finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain
the origin as the only non-vertex lattice point.Comment: 19 page
Mutations of fake weighted projective planes
In previous work by Coates, Galkin, and the authors, the notion of mutation
between lattice polytopes was introduced. Such a mutation gives rise to a
deformation between the corresponding toric varieties. In this paper we study
one-step mutations that correspond to deformations between weighted projective
planes, giving a complete characterisation of such mutations in terms of
T-singularities. We show also that the weights involved satisfy Diophantine
equations, generalising results of Hacking-Prokhorov.Comment: 14 pages, 2 figure
The boundary volume of a lattice polytope
For a d-dimensional convex lattice polytope P, a formula for the boundary
volume is derived in terms of the number of boundary lattice points on the
first \floor{d/2} dilations of P. As an application we give a necessary and
sufficient condition for a polytope to be reflexive, and derive formulae for
the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give
applications to reflexive order polytopes, and to the Birkhoff polytope.Comment: 21 pages; subsumes arXiv:1002.1908 [math.CO]; to appear in the
Bulletin of the Australian Mathematical Societ
Minimality and mutation-equivalence of polygons
We introduce a concept of minimality for Fano polygons. We show that, up to
mutation, there are only finitely many Fano polygons with given singularity
content, and give an algorithm to determine the mutation-equivalence classes of
such polygons. This is a key step in a program to classify orbifold del Pezzo
surfaces using mirror symmetry. As an application, we classify all Fano
polygons such that the corresponding toric surface is qG-deformation-equivalent
to either (i) a smooth surface; or (ii) a surface with only singularities of
type 1/3(1,1).Comment: 29 page
Gorenstein formats, canonical and Calabi-Yau threefolds
We extend the known classification of threefolds of general type that are
complete intersections to various classes of non-complete intersections, and
find other classes of polarised varieties, including Calabi-Yau threefolds with
canonical singularities, that are not complete intersections. Our methods apply
more generally to construct orbifolds described by equations in given
Gorenstein formats.Comment: 25 page
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