🌿Computational Algebraic Geometry Unit 10 – Toric Varieties & Polytopes in Geometry

Key Concepts and Definitions

• Toric varieties algebraic varieties containing algebraic torus as a dense open subset
• Polytopes geometric objects with flat sides, generalizations of polyhedra to arbitrary dimensions
• Fans combinatorial objects consisting of cones, used to construct toric varieties
• Lattices discrete subgroups of a real vector space isomorphic to $\mathbb{Z}^n$
• Cones polyhedral subsets of a real vector space closed under positive scalar multiplication
  • Strongly convex cones contain no line through the origin
  • Rational cones generated by vectors with rational coordinates
• Affine toric varieties defined by a single cone in a lattice
• Projective toric varieties constructed by gluing affine toric varieties along their torus-invariant open subsets
• Torus-invariant divisors divisors on a toric variety invariant under the action of the torus

Geometric Foundations

• Algebraic geometry studies geometric objects defined by polynomial equations
  • Affine varieties subsets of $\mathbb{A}^n$ defined by polynomial equations
  • Projective varieties subsets of $\mathbb{P}^n$ defined by homogeneous polynomial equations
• Toric geometry combines techniques from algebraic geometry, combinatorics, and convex geometry
• Toric varieties have a rich combinatorial structure encoded by fans and polytopes
• Torus actions on toric varieties correspond to combinatorial operations on fans and polytopes
• Orbit-cone correspondence relates orbits of the torus action to cones in the fan
• Toric morphisms equivariant maps between toric varieties, induced by maps of fans or polytopes
• Toric resolution process of resolving singularities of a toric variety by subdividing its fan

Algebraic Structures

• Toric varieties have a natural action of an algebraic torus $(\mathbb{C}^*)^n$
• Character lattice $M$ dual lattice to the lattice $N$ used to define the fan
  • Characters homomorphisms from the torus to $\mathbb{C}^*$
  • Monomials on a toric variety correspond to lattice points in $M$
• Semigroup algebra $\mathbb{C}[S_\sigma]$ associated to each cone $\sigma$ in the fan
  • Semigroup $S_\sigma$ consists of lattice points in the dual cone $\sigma^\vee$
• Cox ring total coordinate ring of a toric variety, graded by the class group
• Torus-equivariant sheaves sheaves on a toric variety equivariant under the torus action
  • Correspond to graded modules over the Cox ring
• Cohomology of line bundles on a toric variety can be computed combinatorially using the polytope

Toric Varieties Explained

• Toric varieties are algebraic varieties containing an algebraic torus as an open dense subset
• Constructed from combinatorial data: fans (affine case) or polytopes (projective case)
• Affine toric variety $U_\sigma$ associated to each strongly convex rational polyhedral cone $\sigma$
  • Defined as the spectrum of the semigroup algebra $\mathbb{C}[S_\sigma]$
• Projective toric variety $X_\Sigma$ associated to a complete fan $\Sigma$
  • Obtained by gluing affine toric varieties $U_\sigma$ for each cone $\sigma \in \Sigma$
• Toric varieties are normal, Cohen-Macaulay, and have a natural torus action
• Orbit-cone correspondence: $k$-dimensional orbits correspond to codimension $k$ cones in the fan
• Torus-invariant divisors correspond to rays (1-dimensional cones) in the fan
• Smooth toric varieties characterized by fans consisting of unimodular cones

Polytopes: Properties and Applications

• Polytopes are geometric objects with flat sides, generalizing polyhedra to higher dimensions
• Lattice polytopes polytopes whose vertices have integer coordinates
  • Correspond to projective toric varieties via the normal fan construction
• Ehrhart theory studies integer points in dilations of a lattice polytope
  • Ehrhart polynomial counts the number of integer points in dilations of a polytope
  • Ehrhart series generating function for the Ehrhart polynomial
• Reflexive polytopes lattice polytopes whose dual polytope is also a lattice polytope
  • Correspond to Gorenstein toric Fano varieties
• Fiber polytopes encode the combinatorics of a linear projection of polytopes
• Polytope algebra graded algebra associated to a lattice polytope, related to toric degenerations
• Mixed volumes of polytopes appear in the Bernstein-Kushnirenko theorem on the number of solutions to a system of polynomial equations

Computational Techniques

• Gröbner bases computational tool for solving polynomial equations and ideal membership problems
  • Used to compute toric ideals and toric degenerations
• Toric ideals prime binomial ideals, vanishing ideals of affine toric varieties
  • Gröbner bases of toric ideals have a combinatorial description
• Toric degenerations flat families of varieties degenerating to a toric variety
  • Useful for studying the geometry and topology of the general fiber
• Triangulations of polytopes related to toric resolutions and Gröbner bases of toric ideals
  • Regular triangulations induced by height functions on the polytope
  • Unimodular triangulations give rise to smooth toric resolutions
• Lattice basis reduction algorithms (LLL) used to find small generators of a lattice
  • Applied to find small toric embeddings and Minkowski bases of polytopes
• Software packages for toric geometry: Macaulay2, Polymake, Sage, Normaliz

Real-world Applications

• Toric varieties appear in various fields of mathematics and beyond
• Algebraic statistics toric models used in the study of discrete statistical models
  • Toric ideals encode independence relations among random variables
• Coding theory toric codes constructed from toric varieties, generalizing Reed-Solomon codes
• Optimization toric varieties used to study linear programming and integer programming problems
  • Polyhedral combinatorics plays a key role in optimization theory
• Phylogenetics toric varieties used to model evolutionary trees and DNA sequence alignment
• Chemical reaction networks toric dynamical systems used to model chemical reactions and population dynamics
• Mirror symmetry toric varieties play a central role in the study of mirror symmetry and Calabi-Yau manifolds
  • Batyrev's construction of mirror pairs using reflexive polytopes

Advanced Topics and Open Problems

• Non-normal toric varieties arising from non-saturated semigroups or non-rational fans
  • Hibi varieties, Rees algebras, and toric Hilbert schemes
• Toric stacks and toric Deligne-Mumford stacks, generalizing toric varieties using stack-theoretic methods
• Toric vector bundles and toric principal bundles, studied using techniques from equivariant topology
• Newton-Okounkov bodies generalizations of Newton polytopes, related to toric degenerations and valuations
• Tropical geometry piecewise-linear version of algebraic geometry, closely related to toric geometry
  • Tropical varieties, tropical compactifications, and tropical intersections
• Ehrhart theory for polytopes: open problems on unimodality, real-rootedness, and positivity of coefficients
• Toric Minimal Model Program: studying the birational geometry of toric varieties using wall-crossing and convex geometry techniques