10.3 Computational aspects of toric varieties

Toric varieties are a fascinating blend of algebra and geometry, connecting abstract math to real-world shapes. They're like the Swiss Army knife of algebraic geometry, useful for tackling all sorts of problems.

In this chapter, we'll dive into the nitty-gritty of building and analyzing toric varieties using computers. We'll explore how to construct them from basic shapes, study their properties, and see how they pop up in other areas of math and science.

Toric Varieties from Combinatorial Data

Constructing Toric Varieties

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• A toric variety is an algebraic variety containing an algebraic torus as a dense open subset, making it a natural generalization of projective spaces (e.g., $\mathbb{P}^n$)
• Toric varieties can be described using combinatorial data such as rational polyhedral fans, which encode the gluing data of affine toric varieties
  • A rational polyhedral fan is a collection of cones in a lattice that satisfies certain properties, such as being closed under taking faces and intersections
• Algorithms for constructing toric varieties from combinatorial data involve computing the dual cone and its associated semigroup algebra
  • The dual cone of a cone $\sigma$ is the set $\sigma^\vee = \{u \in M_\mathbb{R} : \langle u,v \rangle \geq 0 \text{ for all } v \in \sigma\}$, where $M$ is the dual lattice to the lattice $N$ containing the fan
  • The semigroup algebra associated to a cone $\sigma$ is the algebra generated by the lattice points in the dual cone $\sigma^\vee$
• Toric varieties can also be constructed from polytopes, with the normal fan of the polytope giving the combinatorial data of the corresponding toric variety
  • The normal fan of a polytope $P$ is the fan whose cones are the outer normal cones to the faces of $P$

Toric Coordinate Rings and Ideals

• The Cox ring, or total coordinate ring, of a toric variety can be computed from the combinatorial data and encodes the homogeneous coordinates of the toric variety
  • The Cox ring is the polynomial ring generated by the rays of the fan, with one variable per ray
  • The grading on the Cox ring is given by the divisor class group of the toric variety, which can be computed from the fan
• Gröbner basis methods can be used to compute the defining ideal of a toric variety from its combinatorial description
  • The defining ideal of a toric variety is a prime ideal in the Cox ring that encodes the relations between the homogeneous coordinates
  • Gröbner bases provide a computational tool for finding generators of the defining ideal and studying its properties (e.g., Hilbert polynomial, primary decomposition)

Geometry of Toric Varieties

Orbit-Cone Correspondence

• Toric varieties have a rich geometric structure that can be studied using computational methods from algebraic geometry and combinatorics
• The orbit-cone correspondence relates the orbits of the torus action on a toric variety to the cones in its associated fan, providing a dictionary between geometry and combinatorics
  • Each cone $\sigma$ in the fan corresponds to an orbit $O_\sigma$ of the torus action, with dim $O_\sigma$ = codim $\sigma$
  • The closure of an orbit $O_\sigma$ is itself a toric variety, corresponding to the star of the cone $\sigma$ in the fan
• Toric varieties are examples of normal varieties, and their singularities can be studied using the combinatorics of their defining fans
  • A toric variety is smooth if and only if each cone in its fan is unimodular (i.e., generated by a subset of a lattice basis)
  • Singularities of a toric variety correspond to non-simplicial cones in the fan and can be resolved by subdividing the fan

Toric Resolutions and Intersection Theory

• Toric resolutions, obtained by subdividing the fan, can be used to resolve singularities of toric varieties
  • A toric resolution is a morphism of toric varieties that is an isomorphism over the smooth locus and resolves the singularities
  • Toric resolutions can be constructed by subdividing the fan into simplicial cones, which corresponds to a sequence of blowups of the toric variety
• Intersection theory on toric varieties can be studied using the combinatorics of the fan, with the Chow ring and cohomology ring having explicit descriptions in terms of the rays and cones
  • The Chow ring of a toric variety is isomorphic to the Stanley-Reisner ring of the fan, which is a quotient of a polynomial ring by monomials corresponding to non-faces of the fan
  • The cohomology ring of a toric variety can be computed using the Jurkiewicz-Danilov theorem, which gives a combinatorial description in terms of the rays and cones of the fan

Examples and Applications

• Toric varieties provide a rich source of examples in algebraic geometry, such as Fano varieties (e.g., $\mathbb{P}^n$), Calabi-Yau varieties, and varieties with positive Ricci curvature
  • A toric Fano variety is a toric variety whose anticanonical divisor is ample, which can be characterized by the convexity of the support of its fan
  • Toric Calabi-Yau varieties are toric varieties with trivial canonical bundle, which correspond to reflexive polytopes
• Toric varieties have applications to other fields such as mathematical physics, where they arise naturally in the study of mirror symmetry and string theory
  • Mirror symmetry relates the complex geometry of a Calabi-Yau variety to the symplectic geometry of its mirror, which can be constructed as a toric variety
  • Toric geometry provides a computational framework for studying the geometry of Calabi-Yau varieties and their mirrors, with applications to enumerative geometry and physics

Invariants of Toric Varieties

Software Packages

• There are several software packages available for computing invariants of toric varieties, such as Macaulay2, Sage, and Polymake
  • Macaulay2 is a software system for computational algebraic geometry that includes a package for toric varieties
  • Sage is an open-source mathematical software system that includes a module for toric geometry
  • Polymake is a software framework for polyhedral and polytope computations that includes functionality for toric varieties
• These packages can compute the defining ideal, Hilbert polynomial, and Betti numbers of a toric variety from its combinatorial data
  • The defining ideal of a toric variety can be computed using Gröbner basis methods, as implemented in Macaulay2 and Sage
  • The Hilbert polynomial of a toric variety encodes its dimension and degree, and can be computed from the fan using the Ehrhart theory of polytopes
  • The Betti numbers of a toric variety can be computed using the Stanley-Reisner ring of the fan and algebraic techniques such as minimal free resolutions

Intersection Theory and Visualization

• The intersection theory of toric varieties can be studied computationally, with packages able to compute the Chow ring, cohomology ring, and Chern classes
  • The Chow ring and cohomology ring of a toric variety can be computed using the combinatorial descriptions mentioned earlier, as implemented in Sage and Macaulay2
  • The Chern classes of a toric variety can be computed using the combinatorial description of the tangent bundle in terms of the fan
• Software can also be used to study the geometry of polytopes associated to toric varieties, such as computing the volume, lattice points, and Ehrhart polynomial
  • The volume of a polytope can be computed efficiently using triangulations and the Euclidean algorithm
  • The lattice points in a polytope can be enumerated using algorithms from computational geometry and combinatorics
  • The Ehrhart polynomial of a lattice polytope encodes the number of lattice points in dilations of the polytope and can be computed using generating function techniques
• Toric varieties can be visualized using software packages, providing insight into their geometric structure
  • Polymake includes functionality for visualizing polytopes and fans, which can be used to visualize the combinatorial data of toric varieties
  • Sage includes a module for plotting toric varieties and their associated polytopes, providing a way to visualize the geometry of toric varieties

Toric Geometry Applications

Applications to Algebraic Geometry

• Toric geometry provides a bridge between algebraic geometry and combinatorics, allowing techniques from one field to be applied to the other
• Toric varieties can be used to construct explicit examples of algebraic varieties with desired properties, such as Fano varieties or varieties with positive Ricci curvature
  • Toric Fano varieties can be classified using the combinatorics of polytopes, and provide a rich source of examples in the study of Fano varieties
  • Toric varieties with positive Ricci curvature can be constructed using the combinatorics of polytopes, providing examples of Kähler-Einstein metrics and the Calabi conjecture
• The combinatorial structure of toric varieties can be used to study the geometry of more general algebraic varieties, such as through toric degenerations or toric embeddings
  • Toric degenerations are a technique for studying the geometry of a variety by degenerating it to a toric variety, which can be studied using combinatorial methods
  • Toric embeddings are a way of embedding a variety into a toric variety, which can be used to study the geometry of the original variety using toric methods

Applications to Combinatorics and Other Fields

• Toric geometry can be used to study the intersection theory of algebraic varieties, with the combinatorics of fans providing explicit formulas for intersection numbers
  • The intersection numbers of divisors on a toric variety can be computed using the combinatorics of the fan and the Bernstein-Kushnirenko theorem
  • The combinatorics of fans can be used to study the enumerative geometry of curves on toric varieties, such as counting rational curves of given degree
• Problems in combinatorics, such as counting lattice points in polytopes or studying the facial structure of polytopes, can be approached using the geometry of toric varieties
  • The Ehrhart theory of lattice polytopes can be studied using toric varieties, providing a geometric approach to problems in enumerative combinatorics
  • The facial structure of a polytope can be studied using the toric variety associated to its normal fan, providing a connection between the combinatorics of polytopes and the geometry of toric varieties
• Toric geometry has applications to mathematical physics, where toric varieties arise naturally in the study of mirror symmetry and string theory
  • Mirror symmetry relates the complex geometry of a Calabi-Yau variety to the symplectic geometry of its mirror, which can be constructed as a toric variety
  • Toric geometry provides a computational framework for studying the geometry of Calabi-Yau varieties and their mirrors, with applications to enumerative geometry and physics