🌿Algebraic Geometry Unit 9 – Toric Varieties and Polyhedra

Key Concepts and Definitions

• Toric varieties algebraic varieties containing a torus as a dense open subset, allowing techniques from combinatorics and convex geometry to study their properties
• Polyhedra convex hulls of finitely many points in a real vector space, playing a crucial role in the construction and classification of toric varieties
• Fans collections of strongly convex rational polyhedral cones, providing a combinatorial description of toric varieties
• Torus invariant divisors divisors on a toric variety that are invariant under the action of the torus, corresponding to certain combinatorial data of the fan
• Orbit-cone correspondence bijection between orbits of the torus action on a toric variety and cones in the associated fan
  • Allows studying the geometry of toric varieties using combinatorial data
• Homogeneous coordinate ring graded ring associated with a toric variety, reflecting its geometric and combinatorial properties
• Polytopes bounded polyhedra, often used to construct projective toric varieties via their normal fans

Foundations of Toric Geometry

• Toric varieties introduced in the 1970s as a class of algebraic varieties with a torus action, combining techniques from algebraic geometry, combinatorics, and convex geometry
• Torus $(\mathbb{C}^*)^n$ an algebraic group isomorphic to the product of $n$ copies of the multiplicative group of non-zero complex numbers
  • Acts on a toric variety with a dense open orbit
• Torus action on a toric variety can be described using combinatorial data, such as fans and polyhedra
• Affine toric varieties can be constructed as spectra of semigroup algebras associated with lattice cones
• Projective toric varieties obtained as projective closures of affine toric varieties, often described using polytopes
• Toric morphisms equivariant morphisms between toric varieties, induced by morphisms of fans or polyhedra
• Torus invariant subvarieties subvarieties of a toric variety that are invariant under the torus action, corresponding to subfans or faces of polyhedra

Polyhedra and Fans

• Polyhedra fundamental objects in toric geometry, used to construct and study toric varieties
  • Defined as intersections of finitely many halfspaces in a real vector space
• Lattice polyhedra polyhedra whose vertices have integer coordinates with respect to a given lattice
  • Play a key role in the construction of projective toric varieties
• Fans collections of strongly convex rational polyhedral cones satisfying certain conditions
  • Provide a combinatorial description of toric varieties
• Cones subsets of a real vector space closed under taking non-negative linear combinations of their elements
  • Building blocks of fans and polyhedra
• Face lattice partially ordered set of faces of a polyhedron or fan, capturing its combinatorial structure
• Normal fan fan associated with a polytope, obtained by taking the cones over the faces of the polar polytope
  • Used to construct projective toric varieties from polytopes
• Refinement of fans process of subdividing cones in a fan to obtain a new fan, corresponding to a birational morphism of toric varieties

Constructing Toric Varieties

• Toric varieties can be constructed from combinatorial data, such as fans and polyhedra
• Affine toric varieties constructed as spectra of semigroup algebras associated with lattice cones
  • Semigroup algebra $\mathbb{C}[S]$ algebra generated by the elements of a semigroup $S$, with multiplication induced by the semigroup operation
• Projective toric varieties obtained as projective closures of affine toric varieties, often described using polytopes and their normal fans
• Cox construction a general method for constructing toric varieties using homogeneous coordinates and fans
  • Homogeneous coordinate ring $S$ graded ring associated with a fan, with variables corresponding to the rays of the fan and grading determined by the divisor class group
  • Toric variety obtained as the GIT quotient of $\mathbb{C}^n \setminus V(B)$ by the torus $\text{Hom}(\text{Cl}(X), \mathbb{C}^*)$, where $B$ is the irrelevant ideal and $\text{Cl}(X)$ is the divisor class group
• Toric varieties as quotients toric varieties can be realized as quotients of open subsets of affine spaces by subtori
  • Allows studying toric varieties using tools from geometric invariant theory (GIT)
• Toric varieties from polyhedra projective toric varieties can be constructed from lattice polyhedra using their normal fans
  • Vertices of the polyhedron correspond to torus fixed points on the toric variety

Properties of Toric Varieties

• Toric varieties have a rich structure and many desirable properties due to their combinatorial nature
• Orbit decomposition toric varieties can be decomposed into a disjoint union of orbits of the torus action
  • Orbits correspond to cones in the associated fan
• Torus fixed points points on a toric variety that are fixed by the torus action, corresponding to the vertices of the associated polyhedron or the rays of the fan
• Torus invariant divisors divisors on a toric variety that are invariant under the torus action
  • Correspond to certain combinatorial data of the fan or polyhedron (support functions or Cartier data)
• Intersection theory on toric varieties can be studied using combinatorial techniques, such as mixed volumes and Bernstein's theorem
• Cohomology ring of a smooth projective toric variety can be described using the Stanley-Reisner ring of the associated fan
• Toric varieties as Cox rings Cox rings provide an alternative description of toric varieties, generalizing the homogeneous coordinate ring construction
• Toric varieties are normal, Cohen-Macaulay, and have rational singularities

Applications in Algebraic Geometry

• Toric geometry has numerous applications in algebraic geometry and related fields
• Resolution of singularities toric methods can be used to construct explicit resolutions of singularities for certain classes of varieties
  • Achieved by refining the associated fan or subdividing the polyhedron
• Mirror symmetry toric varieties play a crucial role in the study of mirror symmetry, providing a testing ground for various conjectures and constructions
  • Batyrev's construction uses polar dual polytopes to construct mirror pairs of Calabi-Yau varieties
• Gromov-Witten theory toric varieties are important in the study of Gromov-Witten invariants and quantum cohomology
  • Toric methods simplify the computation of these invariants using localization techniques
• Moduli spaces certain moduli spaces in algebraic geometry can be constructed and studied using toric techniques
  • Moduli spaces of stable maps, stable curves, and abelian varieties have toric compactifications
• Tropical geometry toric varieties provide a bridge between algebraic geometry and tropical geometry
  • Tropicalization of a subvariety of a torus can be studied using the associated fan or polyhedron
• Arithmetic geometry toric varieties over finite fields and their zeta functions can be studied using combinatorial techniques
  • Dwork's congruence formula relates the zeta function of a toric variety to the combinatorics of the associated fan

Examples and Exercises

• Projective spaces $\mathbb{P}^n$ are toric varieties associated with the simplex $\Delta^n$
  • Fan consists of cones over the faces of the simplex
• Hirzebruch surfaces $\mathbb{F}_a$ are toric surfaces associated with certain trapezoids
  • Normal fans of these trapezoids give rise to the Hirzebruch surfaces
• Weighted projective spaces $\mathbb{P}(a_0, \ldots, a_n)$ are toric varieties associated with weighted simplices
  • Quotients of $\mathbb{C}^{n+1} \setminus \{0\}$ by the weighted action of $\mathbb{C}^*$
• Toric blow-ups blow-ups of toric varieties at torus fixed points can be described using star subdivisions of the associated fan
• Toric Fano varieties toric varieties with ample anticanonical divisor, corresponding to reflexive polytopes
• Toric Calabi-Yau varieties toric varieties with trivial canonical bundle, associated with reflexive polytopes and their polar duals
• Exercises
  • Construct the fan and polyhedron associated with a given toric variety
  • Compute the homogeneous coordinate ring and Cox ring of a toric variety
  • Determine the torus invariant divisors and their linear equivalence classes
  • Calculate the cohomology ring and intersection numbers on a smooth projective toric variety

Advanced Topics and Open Problems

• Toric varieties over fields other than $\mathbb{C}$, such as finite fields or $p$-adic fields
  • Arithmetic properties and zeta functions of toric varieties over finite fields
• Non-normal toric varieties arising from non-saturated semigroups or non-rational fans
  • Hilbert basis and normalization of toric varieties
• Toric stacks and orbifolds generalizations of toric varieties allowing for quotient singularities
  • Stacky fans and stacky polytopes
• Toric degenerations and toric Gröbner bases
  • Using toric geometry to study degenerations of algebraic varieties and compute Gröbner bases
• Toric varieties in positive characteristic and their Frobenius morphisms
  • Ordinary and non-ordinary toric varieties, Hasse-Weil zeta functions
• Toric varieties and Newton-Okounkov bodies
  • Using valuations and Newton-Okounkov bodies to study toric varieties and their generalizations
• Open problems
  • Classification of smooth projective toric varieties in higher dimensions
  • Combinatorial characterization of toric varieties with certain geometric properties (Fano, Calabi-Yau, etc.)
  • Toric geometry and mirror symmetry beyond the Calabi-Yau case
  • Applications of toric geometry in arithmetic and algebraic geometry, such as the study of rational points and zeta functions