10.1 Introduction to toric varieties
4 min read•july 30, 2024
Toric varieties are a special class of algebraic varieties that combine geometry and combinatorics. They're defined by a and characterized by a , a collection of cones in a lattice. This connection allows us to study complex geometric objects using simpler combinatorial tools.
Toric varieties are incredibly useful in algebraic geometry. They provide concrete examples for abstract concepts and help us understand properties like and singularities. By manipulating the fan, we can construct and modify toric varieties, making them valuable for exploring broader geometric ideas.
Toric Varieties: Definition and Properties
Definition and Characterization
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- Define toric varieties as normal algebraic varieties containing a torus as a dense open subset, with the torus action extending to the entire variety
- Characterize toric varieties by a combinatorial object called a fan, consisting of a collection of cones in a lattice satisfying specific properties
- Equate the dimension of a toric variety to the dimension of the lattice in which its fan is defined
- Classify toric varieties as rational, allowing parameterization by rational functions
Examples and Descriptions
- Provide examples of toric varieties, including projective spaces and weighted projective spaces
- Describe toric varieties using homogeneous coordinates and a set of binomial equations derived from the combinatorial data of the fan
- Illustrate the construction of toric varieties by gluing affine toric varieties corresponding to the cones in the fan
- Explain the role of rays (1-dimensional cones) in determining coordinates and torus-invariant divisors on the toric variety
Constructing Toric Varieties from Combinatorial Data
Fans and Cones
- Define a fan as a collection of strongly convex rational polyhedral cones in a lattice, satisfying the condition that faces of each cone are also in the fan and the intersection of any two cones is a face of both
- Establish the correspondence between cones in the fan and affine toric varieties, with the fan itself corresponding to the toric variety obtained by gluing these affine pieces together
- Highlight the significance of maximal cones in the fan as they correspond to the affine charts covering the toric variety
Construction Process
- Outline the process of constructing a toric variety from a fan by associating a variable to each ray, determining relations among these variables from the combinatorial data of the fan, and gluing the resulting affine pieces together
- Demonstrate the construction process with concrete examples, such as the projective space or the Hirzebruch surface
- Discuss the role of the orbit-cone correspondence in relating the orbits of the torus action on a toric variety to the cones in its fan, providing a geometric interpretation of the combinatorial data
Toric Varieties in Algebraic Geometry
Examples and Counterexamples
- Emphasize the importance of toric varieties as a rich source of examples and counterexamples in algebraic geometry, with many properties translating into combinatorial conditions on the fan
- Provide specific instances where toric varieties serve as illustrative examples or counterexamples for concepts in algebraic geometry, such as the existence of non-projective varieties or the behavior of the Picard group
Applications and Connections
- Explore the use of toric varieties in studying the behavior of algebraic varieties under degenerations or specializations, through the lens of toric degenerations
- Discuss the computation of invariants of toric varieties, such as the Picard group, the cone of effective divisors, and cohomology groups, directly from the fan
- Highlight the role of toric varieties in the study of mirror symmetry, as they provide a setting for constructing mirror pairs and investigating the relationship between the complex and symplectic geometry of Calabi-Yau manifolds
Singularities and Smoothness of Toric Varieties
Characterization of Singularities
- Characterize the smoothness of a toric variety in terms of its fan, stating that a toric variety is smooth if and only if each cone in its fan is generated by a subset of a basis of the lattice
- Describe singularities in toric varieties combinatorially using the cones in the fan, defining a singular cone as one that is not generated by a subset of a basis of the lattice
- Explain the local structure of singularities in affine toric varieties corresponding to singular cones, which resemble quotients of affine space by finite abelian groups
Resolution of Singularities
- Discuss the process of resolving singularities in toric varieties by subdividing the cones in the fan until all cones are smooth, known as a toric
- Highlight the non-uniqueness of toric resolutions and the relationship between different resolutions through birational transformations called flips and flops
- Illustrate the resolution process with examples, such as the resolution of the conifold singularity or the resolution of singularities in weighted projective spaces
Applications in the Minimal Model Program
- Explore the use of toric varieties in constructing examples of varieties with specific types of singularities, such as canonical or terminal singularities, which play a role in the minimal model program
- Discuss the implications of the toric description of singularities for the study of birational geometry and the classification of algebraic varieties
- Provide examples of how toric methods have been used to resolve questions or conjectures in the minimal model program, such as the existence of flips or the termination of certain birational processes
Key Terms to Review (19)
Batyrev's Theorem: Batyrev's Theorem states that for a smooth projective variety defined by a toric fan, the number of rational points on the variety can be computed through its associated polytope. This theorem is significant as it connects algebraic geometry and combinatorial geometry, highlighting how properties of toric varieties can be interpreted through the geometry of polytopes. It provides a powerful tool for studying the arithmetic and geometric properties of these varieties.
Complete Toric Variety: A complete toric variety is a specific type of toric variety that is projective and can be realized as a compactification of a torus. It is defined by the geometry of a fan, where each cone corresponds to an affine chart of the variety. This relationship connects it to important concepts like polyhedral geometry, combinatorial data, and algebraic varieties, making it a fundamental object in algebraic geometry.
Cox Ring: The Cox ring is a specific kind of graded ring associated with a toric variety, capturing the algebraic and geometric structure of the variety in a unified way. This ring is built from the homogeneous coordinate rings of the affine patches of a toric variety and plays a crucial role in studying the properties and computations related to these varieties, including their intersection theory and their relations to polytopes.
Deformation Theory: Deformation theory studies how mathematical objects change under small perturbations, focusing on the structure of moduli spaces. This theory is crucial in understanding how various geometric structures, like toric varieties, can be continuously transformed while preserving their essential characteristics. By examining these transformations, one can gain insights into the properties of algebraic varieties and their connections to other areas in geometry and algebra.
Embeddings: Embeddings are a mathematical concept used to map one set of mathematical objects into another space while preserving certain structures and properties. In the context of toric varieties, embeddings allow for the representation of algebraic varieties as subsets of projective spaces, facilitating their study through combinatorial and geometric methods.
Fan: In mathematics, a fan is a collection of cones that are used to describe the combinatorial structure of toric varieties. Each cone represents a direction in a multi-dimensional space, and the way these cones intersect defines the geometry of the associated toric variety. Fans provide a systematic way to study the properties of toric varieties, connecting algebraic geometry and combinatorial geometry through their underlying polytopes and facilitating computations in toric geometry.
Fano Varieties: Fano varieties are a special class of algebraic varieties characterized by having ample anticanonical bundles. This property makes them important in the study of algebraic geometry because they exhibit many favorable geometric and topological properties, such as being non-singular and having rich structure. Fano varieties often serve as important examples in various contexts, including toric geometry, where their combinatorial nature can be analyzed using toric techniques.
Gian-Carlo Rota: Gian-Carlo Rota was a prominent mathematician known for his work in combinatorics, algebraic geometry, and the theory of posets. His contributions to the study of toric varieties and the interplay between ideals and varieties are significant, influencing how we understand the geometric aspects of algebraic structures and their applications in computational algebraic geometry.
Gorenstein: Gorenstein refers to a specific type of commutative ring or algebraic variety that has nice duality properties and is characterized by having a canonical module that is finitely generated and invertible. This property is particularly important in algebraic geometry and commutative algebra, as it ensures that certain cohomological dimensions are well-behaved and provides insight into the structure of singularities within varieties, especially in the context of toric varieties.
Lattice Polytope: A lattice polytope is a convex polytope whose vertices all have integer coordinates, meaning they lie in the integer lattice in some Euclidean space. These structures are crucial in understanding combinatorial and geometric properties of toric varieties, as they help establish connections between geometry and algebraic properties through their associated fan structures and homogeneous coordinate rings.
Morphisms: Morphisms are structure-preserving maps between mathematical objects that allow for the study of relationships and transformations within algebraic structures. They serve as a foundational concept in category theory and provide a way to compare different structures, such as varieties or algebraic objects, by relating them through these maps. Understanding morphisms is essential for exploring the properties of toric varieties and their geometric interpretations.
Normalization: Normalization is the process of transforming a singular algebraic variety into a regular one, ensuring that the resulting space is 'well-behaved' and free from singularities. This process helps in understanding the geometry and structure of varieties, allowing mathematicians to analyze them more effectively, especially in the context of toric varieties where combinatorial data plays a crucial role.
Polytope: A polytope is a geometric object with flat sides, which exists in any number of dimensions. In two dimensions, it is known as a polygon, while in three dimensions, it is called a polyhedron. Polytopes serve as fundamental constructs in various areas of mathematics, including algebraic geometry, particularly in the study of toric varieties where they encode combinatorial data and provide a bridge between geometry and algebra.
Projective Toric Variety: A projective toric variety is a specific type of algebraic variety that is defined by combinatorial data from a convex polytope and can be embedded into projective space. These varieties provide a bridge between algebraic geometry and combinatorial geometry, allowing for the study of geometric properties using the underlying combinatorial structure of polytopes.
Resolution of Singularities: Resolution of singularities is a process in algebraic geometry that aims to replace a singular variety with a non-singular variety, effectively 'resolving' the points of singularity. This technique is crucial for understanding the geometric properties of algebraic varieties and plays an important role in various applications, including toric varieties, where singularities may arise from combinatorial data related to polyhedra.
Smoothness: Smoothness refers to a property of a geometric object where it has no singularities, meaning that it can be described by differentiable functions at every point. In algebraic geometry, smoothness indicates that the local structure of the variety behaves nicely, allowing for well-defined tangent spaces. This concept is crucial in understanding various algebraic constructs, as it ensures that methods applied to these objects, such as intersection theory or homotopy methods, are valid and effective.
Torus Action: A torus action refers to a group action on a geometric object by a torus, which is typically a product of circles, denoted as $T^n$. This action helps in understanding the symmetries and properties of varieties, especially in the realm of toric varieties, where the geometry aligns closely with combinatorial data. By studying torus actions, one can gain insights into the structure and classification of algebraic varieties, particularly in relation to their embedding in projective spaces.
Tropicalization: Tropicalization is a mathematical process that transforms algebraic varieties into a piecewise linear setting by replacing the usual operations of addition and multiplication with maximum and addition, respectively. This transformation allows complex geometric problems to be simplified and analyzed using combinatorial techniques, linking classical algebraic geometry with tropical geometry. The concept plays a vital role in understanding toric varieties and finding applications in various fields, including optimization and computational geometry.
William Fulton: William Fulton is a prominent mathematician known for his contributions to algebraic geometry, particularly in the area of toric varieties and intersection theory. His work has provided significant insights into the geometric structures arising from combinatorial data, particularly how these structures interact with intersection multiplicities and degrees in algebraic varieties.