11.1 Toric varieties and polytopes
8 min read•august 20, 2024
Toric varieties are algebraic varieties containing a torus as a dense open subset. They're built from combinatorial data like fans and polytopes, making them accessible to study using discrete methods. This connection to combinatorics provides a rich class of examples in algebraic geometry.
Polytopes and fans encode the geometry of toric varieties. Lattice polytopes give rise to projective toric varieties, while fans describe affine toric varieties. The interplay between polytope combinatorics and toric variety geometry is central to toric geometry.
Definition of toric varieties
- Toric varieties are algebraic varieties that contain a torus as a dense open subset and the action of the torus on itself extends to an action on the entire variety
- Toric varieties provide a rich class of examples in algebraic geometry and have connections to combinatorics, convex geometry, and representation theory
- Toric varieties can be constructed from combinatorial data such as fans and polytopes, making them accessible to study using discrete methods
Toric varieties from fans
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- A is a collection of cones in a real vector space that satisfies certain compatibility conditions
- Each cone in a fan corresponds to an , and the fan encodes how these affine pieces glue together to form the toric variety
- The rays (1-dimensional cones) in a fan correspond to the torus-invariant divisors on the toric variety
Toric varieties from polytopes
- A polytope is a convex hull of a finite set of points in a real vector space
- The normal fan of a polytope is a fan that encodes the combinatorial structure of the polytope
- The toric variety associated to a polytope is the same as the toric variety associated to its normal fan
Affine toric varieties
- An affine toric variety is a toric variety that can be described as the spectrum of a semigroup algebra
- Each cone in a fan corresponds to an affine toric variety, which is the spectrum of the semigroup algebra of the dual cone
- Affine toric varieties are the building blocks of general toric varieties
Projective toric varieties
- A is a toric variety that admits a torus-equivariant embedding into a projective space
- Projective toric varieties can be constructed from polytopes: the toric variety associated to a lattice polytope is projective
- Examples of projective toric varieties include projective spaces, products of projective spaces, and weighted projective spaces
Polytopes and toric geometry
- Polytopes and fans are the combinatorial objects that encode the geometry of toric varieties
- The interplay between the combinatorics of polytopes and the geometry of toric varieties is a central theme in toric geometry
Lattice polytopes
- A lattice polytope is a polytope whose vertices have integer coordinates
- Lattice polytopes are the polytopes that give rise to projective toric varieties
- The lattice points in a lattice polytope correspond to a basis of the space of global sections of an ample line bundle on the associated toric variety
Normal fan of a polytope
- The normal fan of a polytope is a fan that encodes the combinatorial structure of the polytope
- The cones in the normal fan correspond to the faces of the polytope
- The normal fan of a polytope determines the toric variety associated to the polytope
Polytopes vs fans
- Polytopes and fans are dual objects: the normal fan of a polytope encodes the same combinatorial data as the polytope itself
- Some toric varieties (e.g., affine toric varieties) are more naturally described using fans, while others (e.g., projective toric varieties) are more naturally described using polytopes
- The choice of whether to work with polytopes or fans often depends on the specific problem or context
Moment polytopes
- The moment polytope of a projective toric variety is the image of the variety under the moment map associated to the
- The moment polytope of a projective toric variety is a lattice polytope that encodes the same data as the fan of the variety
- Moment polytopes are a useful tool for studying the geometry and topology of toric varieties, such as their cohomology and intersection theory
Orbits and torus action
- The torus action on a toric variety determines its structure and properties
- Understanding the orbits of the torus action and their closures is crucial for studying toric varieties
Torus orbits in toric varieties
- The torus acts on a toric variety with finitely many orbits
- There is a one-to-one correspondence between the orbits of the torus action and the cones in the fan of the toric variety
- The dimension of an orbit is equal to the codimension of the corresponding cone in the fan
Orbit closures and faces
- The closure of a torus orbit in a toric variety is a toric subvariety
- The orbit closures correspond to the faces of the polytope associated to the toric variety
- The inclusion relations between orbit closures are determined by the face relations in the polytope or fan
Torus invariant divisors
- A torus invariant divisor on a toric variety is a divisor that is invariant under the torus action
- Torus invariant divisors correspond to the rays (1-dimensional cones) in the fan of the toric variety
- The torus invariant divisors generate the Picard group (group of line bundles) of the toric variety
Torus equivariant morphisms
- A torus equivariant morphism between toric varieties is a morphism that commutes with the torus actions on the varieties
- Torus equivariant morphisms correspond to morphisms of fans or polytopes that respect the torus action
- Torus equivariant morphisms are a key tool for studying maps between toric varieties and their properties
Toric resolution of singularities
- Toric varieties can have singularities, but these singularities can be resolved using toric methods
- Toric resolution of singularities is a powerful technique that has applications beyond toric varieties themselves
Cones and affine toric varieties
- Affine toric varieties are determined by cones in the fan of the toric variety
- Singularities of affine toric varieties correspond to non-smooth cones in the fan
- Resolving the singularities of an affine toric variety amounts to subdividing the corresponding cone into smooth cones
Refinement of fans
- A refinement of a fan is a fan that subdivides the cones of the original fan into smaller cones
- Refinements of fans correspond to toric birational morphisms that resolve singularities
- A toric variety is smooth if and only if its fan consists of smooth cones
Toric resolution of toric varieties
- Any toric variety can be resolved by a smooth toric variety via a toric birational morphism
- The resolution can be obtained by refining the fan of the original toric variety into a smooth fan
- Toric resolutions are not unique, but there exists a minimal resolution that is unique up to isomorphism
Resolution of singularities
- Toric resolution of singularities is a special case of the general problem of resolution of singularities in algebraic geometry
- Toric methods provide a constructive approach to resolution of singularities for a large class of algebraic varieties
- Toric resolution has applications to the study of singularities, birational geometry, and the minimal model program
Cohomology of toric varieties
- The cohomology of toric varieties can be studied using combinatorial techniques
- The cohomology rings of toric varieties have a rich structure that reflects the combinatorics of the associated polytopes or fans
Cohomology ring of smooth projective toric varieties
- The cohomology ring of a smooth projective toric variety is isomorphic to the Stanley-Reisner ring of the associated fan
- The Stanley-Reisner ring is a quotient of a polynomial ring by an ideal determined by the combinatorial structure of the fan
- The cohomology ring is generated by the classes of torus invariant divisors, with relations coming from the linear dependence of divisors
Intersection theory on toric varieties
- Intersection theory on toric varieties can be studied using the combinatorics of polytopes
- The intersection numbers of torus invariant divisors can be computed using the mixed volume of the corresponding polytopes
- The intersection theory of toric varieties is closely related to the theory of mixed subdivisions and mixed Hodge structures
Chow rings and polytopes
- The Chow ring of a toric variety is a ring that encodes the intersection theory of algebraic cycles on the variety
- For a smooth projective toric variety, the Chow ring is isomorphic to the cohomology ring and can be described using the associated polytope
- The Chow ring of a singular toric variety can be studied using the combinatorics of the associated fan and its subdivisions
Toric varieties over finite fields
- Toric varieties can be defined over any field, including finite fields
- The cohomology and intersection theory of toric varieties over finite fields have arithmetic analogues that involve counting points
- Toric varieties over finite fields have applications to coding theory, cryptography, and the study of zeta functions of varieties
Toric degenerations
- Toric degenerations are a way of approximating a general algebraic variety by a toric variety
- Toric degenerations have applications to the study of moduli spaces, enumerative geometry, and tropical geometry
Toric degenerations of projective varieties
- A toric degeneration of a projective variety is a flat family of varieties that specializes to a toric variety
- Toric degenerations can be constructed using Gröbner bases and initial ideals of the defining equations of the variety
- The special fiber of a toric degeneration encodes information about the original variety, such as its intersection theory and cohomology
Gröbner bases and initial ideals
- A Gröbner basis is a special generating set of an ideal in a polynomial ring that depends on a choice of monomial order
- The initial ideal of an ideal with respect to a monomial order is the ideal generated by the initial terms of the elements of the ideal
- Gröbner bases and initial ideals are key tools for studying toric degenerations and their properties
SAGBI bases and toric ideals
- A SAGBI (Subalgebra Analogue of Gröbner Basis for Ideals) basis is a special generating set of a subalgebra of a polynomial ring that depends on a choice of monomial order
- Toric ideals are the defining ideals of affine toric varieties and can be studied using SAGBI bases
- SAGBI bases and toric ideals have applications to the study of toric degenerations and their relation to tropical geometry
Tropical geometry and toric degenerations
- Tropical geometry is a piecewise-linear analogue of algebraic geometry that arises as a limit of toric degenerations
- The of an algebraic variety is a piecewise-linear object that encodes information about the toric degenerations of the variety
- Tropical geometry provides a new perspective on the study of algebraic varieties and their moduli spaces, and has connections to combinatorics, topology, and mathematical physics
Key Terms to Review (18)
Affine toric variety: An affine toric variety is a specific type of algebraic variety that can be constructed from a rational polytope and is associated with a torus acting on it. These varieties arise from the combinatorial data of fans and polytopes, where the points in the polytope correspond to monomials in a polynomial ring, giving a geometric interpretation to algebraic concepts. Affine toric varieties serve as a bridge between algebraic geometry and combinatorial geometry, allowing for the exploration of geometric properties through the lens of polytopes.
Deformation: Deformation refers to the process of transforming a geometric object into another shape while preserving certain structural properties. In the context of tropical geometry, deformation plays a critical role in understanding how various algebraic structures can change and adapt, affecting key concepts such as tropical discriminants, the structure of varieties, and relationships between polytopes.
Fan: In the context of toric varieties and polytopes, a fan is a collection of cones that represents a way to construct a toric variety from combinatorial data. Each cone in a fan corresponds to a cone in the polytope, and the fans help describe how these cones intersect and combine to form complex geometric structures. Fans are crucial for understanding the relationship between geometry and algebraic varieties, serving as a bridge between these two areas.
Giorgio Ottaviani: Giorgio Ottaviani is an influential mathematician known for his significant contributions to the field of algebraic geometry, particularly in tropical geometry. His work focuses on tropical polynomial functions and their applications, exploring the interplay between algebraic and combinatorial structures in mathematics.
Grassmannians: Grassmannians are geometric spaces that parameterize all linear subspaces of a given dimension within a vector space. They play a crucial role in various fields such as algebraic geometry and topology, particularly in understanding toric varieties and polytopes, where the connections between combinatorial data and geometric structures become essential.
Minkowski-Weyl Theorem: The Minkowski-Weyl Theorem provides a powerful connection between convex polytopes and their corresponding toric varieties by stating that any convex polytope can be represented in terms of its vertices and its supporting hyperplanes. This theorem is essential for understanding how polytopes relate to algebraic geometry, particularly in the study of toric varieties, which are geometric objects defined by combinatorial data from these polytopes.
Multinomial varieties: Multinomial varieties are geometric objects defined by the vanishing of multinomial functions, which are polynomials with multiple variables, each raised to a non-negative integer power. These varieties arise naturally in the study of toric varieties and polytopes, where the combinatorial structure of polytopes can be linked to algebraic properties through their associated coordinate rings. They provide a rich framework for understanding how geometry and algebra interact, particularly in relation to the monomial and toric ideals derived from these functions.
Newton Polytope: A Newton polytope is a convex hull of the points corresponding to the exponents of the monomials in a polynomial, essentially representing the geometric shape formed by those exponents. It plays a crucial role in understanding tropical geometry, as it helps to analyze the behavior of polynomials under tropicalization and influences the structure of tropical hypersurfaces, cycles, and Hodge theory.
Polytopal Complex: A polytopal complex is a collection of polyhedra that are glued together along their faces in a consistent manner, creating a combinatorial structure that encodes the geometric and topological properties of the shapes involved. These complexes are crucial in understanding the relationship between geometry and algebraic structures, particularly in the study of toric varieties where polytopes play a significant role in defining the geometry of algebraic varieties.
Projective toric variety: A projective toric variety is a type of algebraic variety that is constructed from a combinatorial object known as a polytope. These varieties are defined by their relation to fans, which are collections of cones in a vector space that encode the combinatorial and geometric data necessary to study the variety. Projective toric varieties provide a way to understand how geometry interacts with combinatorics through the lens of polytopes and their associated torus actions.
Supporting Hyperplane: A supporting hyperplane is a hyperplane that touches a convex set at least at one point and does not intersect the interior of that set. This concept is crucial in understanding the geometry of polytopes and toric varieties, as it helps describe how these geometric objects can be analyzed through their vertices and facets. Supporting hyperplanes play a significant role in determining the face structure of polytopes, connecting the algebraic properties of toric varieties with their geometric representations.
Toric Birkhoff Theorem: The Toric Birkhoff Theorem establishes a connection between toric varieties and the combinatorial properties of polytopes, particularly focusing on the notion of equivariant embeddings. It provides a framework for understanding how certain algebraic and geometric structures can be derived from the combinatorial data of a polytope, linking algebraic geometry with polyhedral geometry.
Torus action: A torus action is a continuous group action by a torus, typically denoted as $(\mathbb{T}^n)$, on a topological space or algebraic variety. This action introduces a way to analyze the structure of the variety through the symmetries provided by the torus, revealing important geometric and combinatorial properties, especially in relation to toric varieties and polytopes.
Tropical Convexity: Tropical convexity refers to a geometric structure that arises in tropical geometry, where the classical notions of convex sets and convex hulls are redefined using the tropical semiring. This concept allows for the study of combinatorial and algebraic properties of sets defined over the tropical numbers, enhancing our understanding of tropical equations, hypersurfaces, and halfspaces.
Tropical Linearity: Tropical linearity refers to a concept in tropical geometry where the usual operations of addition and multiplication are replaced with their tropical counterparts. In this framework, addition is interpreted as taking the maximum of two values, while multiplication is treated as ordinary addition. This new perspective allows us to analyze geometric structures like hypersurfaces, polytopes, and halfspaces in a different light, highlighting the rich combinatorial properties and connections between various mathematical concepts.
Tropical Polytope: A tropical polytope is a geometric object defined in tropical geometry, which is a piecewise-linear analogue of classical polytopes. It is formed by taking the convex hull of a set of points in tropical space, where the operations of addition and multiplication are replaced by minimum and addition, respectively, allowing for a new way to study combinatorial structures and optimization problems.
Tropicalization: Tropicalization is the process of translating algebraic varieties and their properties into a piecewise-linear setting using tropical geometry. This allows for the study of complex geometric structures through combinatorial means, enabling a more accessible approach to problems involving algebraic curves and surfaces.
Velasco: Velasco is a significant figure in tropical geometry, known for his contributions to tropical Hodge theory and the study of toric varieties. His work provides important insights into the relationships between algebraic geometry, combinatorics, and the geometry of numbers. Through the lens of Velasco's research, one can explore how tropical techniques yield new understandings of classical algebraic concepts.