Interactions with Lattice Polytopes: Magdeburg, Germany, September 2017 (Springer Proceedings in Mathematics & Statistics, 386)

Preface
Contents
Contributors
1 Difference Between Families of Weakly and Strongly Maximal Integral Lattice-Free Polytopes
	1.1 Introduction
	1.2 An Approach to Construction of Polytopes in mathcalLd  mathcalMd
	1.3 Lattice-Free Axis-Aligned Simplices
	1.4 Proof of the Main Result
	References
2 On the Fine Interior of Three-Dimensional Canonical Fano Polytopes
	2.1 Introduction
	2.2 Almost Reflexive Polytopes of Dimension 3 and 4
	2.3 Canonical Fano 3-Topes with Δ`3́9`42`\"̇613A``45`47`\"603AFI = {0}
	2.4 Asymmetric Fine Interior of Dimension 1
	2.5 Symmetric Fine Interior of Dimension 1
	2.6 Fine Interior of Dimension 3
	2.7 Hollow 3-Topes with Non-empty Fine Interior
	References
3 Lattice Distances in 3-Dimensional Quantum Jumps
	3.1 Introduction
	3.2 Distances in 3-Dimensional Quantum Jumps or Unions
		3.2.1 Quantum Jumps (Q,x) with Q of Dimension 2
		3.2.2 Quantum Unions of Lattice Segments
		3.2.3 Quantum Jumps (Q,x) with Q of Dimension 3
	3.3 Distance from the Boundary to the Inner Polytope
		3.3.1 Inner Polytope of Dimension 1
		3.3.2 Inner Polytope of Dimension 2
		3.3.3 Inner Polytope of Dimension 3
	References
4 Flag Matroids: Algebra and Geometry
	4.1 Introduction
	4.2 Matroids: Combinatorics
		4.2.1 Introduction to Matroids
		4.2.2 The Tutte Polynomial
		4.2.3 The Base Polytope
		4.2.4 Definition via Gale Orderings
		4.2.5 The Matroid Union Theorem
	4.3 Polymatroids: Combinatorics
		4.3.1 The Tutte Polynomial for Polymatroids
	4.4 Flag Varieties: Geometry
		4.4.1 Representations and Characters
		4.4.2 Grassmannians
		4.4.3 Flag Varieties
	4.5 Representable Matroids: Combinatorics and Geometry
	4.6 Introduction to Flag Matroids
		4.6.1 Flag Matroids: Definition
		4.6.2 Matroid Quotients
		4.6.3 Representable Flag Matroids
		4.6.4 Flag Matroid Polytopes
		4.6.5 Flag Matroids and Torus Orbits
	4.7 Representable Polymatroids
		4.7.1 Comparison Between Polymatroids and Flag Matroids
	4.8 Equivariant K-theory
		4.8.1 A Very Brief Introduction to K-theory
		4.8.2 Explicit Construction via Equivariant Localisation
		4.8.3 A Short Review on Cones and Their Hilbert Series
		4.8.4 Matroids and the K-theory of Grassmannians
		4.8.5 Flag Matroids and the K-theory of Flag Varieties
		4.8.6 The Tutte Polynomial via K-theory
	4.9 Open Problems
	References
5 Classification of Minimal Polygons with Specified Singularity Content
	5.1 Introduction
	5.2 Mutations of Fano Polygons and Singularity Content
		5.2.1 Mutations
		5.2.2 Singularity Content
		5.2.3 Hirzebruch–Jung Continued Fractions and Applications to Algebraic Geometry
	5.3 Minimal Fano Polygons
	5.4 Algorithm to Calculate Minimal Polygons with Given Basket
		5.4.1 Special Facets
		5.4.2 Description of Algorithm
	5.5 Minimal Fano Polygons with B= { m1 times13(1,1) , m2 times16(1,1) }
	5.6 Minimal Fano Polygons with  B= { m times15(1,1) }
	References
6 On the Topology of Fano Smoothings
	6.1 Introduction
	6.2 Cohomology and Vanishing Cycles
		6.2.1 The Cohomology of Toric Varieties
		6.2.2 The Vanishing Cycle Exact Sequence
	6.3 Smoothing Toric Fano Threefolds
		6.3.1 Computing the Betti Numbers of the Smoothing
		6.3.2 Minkowski Polynomials and Smoothings
	6.4 Examples
		6.4.1 Cube
		6.4.2 A Singular Toric Variety with Two Different Smoothings
		6.4.3 An Example with Transverse A2 Singularities
	6.5 Betti Numbers Depend Only on the Mirror Laurent Polynomial
		6.5.1 Type I Moves
		6.5.2 Type II Moves
		6.5.3 Proof of Theorem 9
	6.6 Systematic Analysis
	References
7 Computing Seshadri Constants on Smooth Toric Surfaces
	7.1 Introduction
		7.1.1 Open Questions and Future Directions
	7.2 Background
		7.2.1 Toric Geometry
		7.2.2 Adjunction Theory for Toric Varieties
	7.3 Seshadri Constants and Jet Separation
	7.4 Seshadri Constants and Unnormalized Spectral Values
	7.5 Characterizing Polygons Whose Core is a Point
	References
8 The Characterisation Problem of Ehrhart Polynomials of Lattice Polytopes
	8.1 Introduction
	8.2 Preliminaries
	8.3 Small Dimensions
	8.4 Small Volumes
	8.5 Palindromic
	8.6 Small Degrees
	8.7 Universal Inequalities
	References
9 The Ring of Conditions for Horospherical Homogeneous Spaces
	9.1 Motivation
	9.2 Linear Algebraic Groups: A Crash Course
	9.3 Spherical Varieties
		9.3.1 The Luna–Vust Theory of Spherical Embeddings
		9.3.2 The Classification of Spherical Homogeneous Spaces
		9.3.3 The Complete Picture in the Horospherical Case
	9.4 The Ring of Conditions of a Horospherical Variety
		9.4.1 The Horospherical Case
	References
10 Linear Recursions for Integer Point Transforms
	10.1 Introduction
	10.2 Characteristic Functions and Valuations
	10.3 A Multivariate Recursion
	10.4 Brion\'s Theorem
	10.5 Schur Polynomials
	References
11 Schubert Calculus on Newton–Okounkov Polytopes
	11.1 Introduction
	11.2 Preliminaries
		11.2.1 Polytope Ring
		11.2.2 GZ Polytopes in Types B and C
		11.2.3 Newton–Okounkov Polytopes of Flag Varieties
	11.3 Geometric Mitosis
		11.3.1 Type A: GZ Polytopes
		11.3.2 Type C2-3: DDO Polytopes
		11.3.3 Type C: GZ Polytopes
		11.3.4 Type B: GZ Polytopes
	References
12 An Eisenbud–Goto-Type Upper Bound for the Castelnuovo–Mumford Regularity of Fake Weighted Projective Spaces
	12.1 Introduction
	12.2 Background Material
		12.2.1 Toric Varieties and Lattice Simplices
		12.2.2 Ehrhart Theory
	12.3 k-Normality of Very Ample Simplices
	12.4 Eisenbud–Goto-Type Upper Bound for Very Ample Simplices
	12.5 Eisenbud–Goto Conjecture for Non-hollow Very Ample Simplices
	12.6 Final Remarks
		12.6.1 Hollow Very Ample Simplices
	References
13 Toric Degenerations in Symplectic Geometry
	13.1 Introduction
	13.2 Toric Degenerations
	13.3 Gromov Width
		13.3.1 Results About Coadjoint Orbits
		13.3.2 A Sketch of the Proof of Theorem 1
	13.4 Cohomological Rigidity
		13.4.1 Toric Degenerations for Symplectic Toric Manifolds
		13.4.2 Cohomological Rigidity for Bott Manifolds
	References
14 On Deformations of Toric Fano Varieties
	14.1 Introduction
		14.1.1 Outline
		14.1.2 Notation and Conventions
	14.2 Deformations
		14.2.1 Infinitesimal Deformations
		14.2.2 Smoothings
		14.2.3 Invariants
	14.3 Deformations of Affine Toric Varieties
		14.3.1 Toric Singularities
		14.3.2 The Affine Cone over the Del Pezzo Surface  of Degree 7
	14.4 Deformations of Toric Fano Varieties
		14.4.1 Fano Polytopes
		14.4.2 Two Sufficient Conditions for Non-smoothability
		14.4.3 Rigidity
		14.4.4 Toric del Pezzo Surfaces
		14.4.5 Toric Fano Threefolds with Isolated Singularities
		14.4.6 The Projective Cone over the Del Pezzo Surface of Degree 7
		14.4.7 Another Sufficient Condition for Non-smoothability
		14.4.8 Other Methods
	14.5 Lists of Reflexive Polytopes of Dimension 3
	References
15 Polygons of Finite Mutation Type
	15.1 Introduction
	15.2 Quivers and Cluster Algebras
	15.3 Mutations of Polytopes
	15.4 Finite Type Classification
	References
16 Orbit Spaces of Maximal Torus Actions on Oriented Grassmannians of Planes
	16.1 Introduction
	16.2 TmathbbC-Varieties and Their T-Orbit Spaces
	16.3 Oriented Grassmanians of Planes as TmathbbC-Varieties
	16.4 Complexity-One TmathbbC-Varieties
	References
17 The Reflexive Dimension of (0,1)-Polytopes
	17.1 Introduction
	17.2 Toric Ideals and Reflexive Polytopes
	17.3 Reflexive Polytopes Arising from Order Polytopes and Chain Polytopes
	17.4 Reflexive Polytopes Arising from the Stable Set Polytopes of Perfect Graphs
	17.5 Reflexive Polytopes Arising from Edge Polytopes
	References