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"Polytopes"
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Existence of unimodular triangulations — positive results
Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course,
combinatorics.
In this article, we review several classes of polytopes that do have unimodular triangulations and constructions
that preserve their existence.
We include, in particular, the first effective proof of the classical result by
Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an
explicit (although doubly exponential) bound for the dilation factor.
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FINITENESS OF LOG MINIMAL MODELS AND NEF CURVES ON -FOLDS IN CHARACTERISTIC
In this article, we prove a finiteness result on the number of log minimal models for 3-folds in$\\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic$p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor$K_{X}$of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on$X$.
Journal Article
Proof of a Conjecture of Batyrev and Juny on Gorenstein Polytopes
A d-dimensional lattice polytope P is Gorenstein if it has a multiple rP that is a reflexive polytope up to translation by a lattice vector. The difference d+1-r is called the degree of P. We show that a Gorenstein polytope is a lattice pyramid if its dimension is at least three times its degree. This was previously conjectured by Batyrev and Juny. We also present a refined conjecture and prove it for IDP Gorenstein polytopes.
Journal Article
The h∗-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their γ-Positivity
A lattice polytope P⊂Rd is called a locally anti-blocking polytope if for any closed orthant Rεd in Rd, P∩Rεd is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the h∗-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the γ-positivity of h∗-polynomials of locally anti-blocking reflexive polytopes.
Journal Article
Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements,
interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the
monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained.
eBook
Geometrical structures of nested polyhedra
The polyhedra with A 3 , B 3 / C 3 , H 3 reflection symmetry group G in the real 3 D space are considered. The recursive rules for finding orbits with smaller radii, which provide the structures of nested polytopes, are demonstrated.
Journal Article
Monotone Paths on Cross-Polytopes
In the early 1990s, Billera and Sturmfels introduced the monotone path polytope (MPP), an important case of the general theory of fiber polytopes, which has led to remarkable combinatorics. Given a pair (P,φ) of a polytope P and a linear functional φ, the MPP is obtained from averaging the fibers of the projection φ(P). They also showed that MPPs of (regular) simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. As a natural follow-up we investigate the monotone paths of cross-polytopes for a generic linear functional φ. We show the face lattice of the MPP of the cross-polytope is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We also describe its f-vector, some geometric realizations, an irredundant inequality description, the 1-skeleton and we compute its diameter. In contrast to the case of simplices and hyper-cubes, monotone paths on cross-polytopes are not always coherent.
Journal Article
Abstract Regular Polytopes
Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.
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