9.4 Toric resolutions and singularities
4 min read•august 14, 2024
Toric resolutions are powerful tools for fixing singularities in toric varieties. By tweaking the , we can smooth out rough spots in these geometric objects. It's like using a magic eraser on a bumpy surface!
This process bridges the gap between combinatorics and algebraic geometry. By studying how fans change, we gain insights into the nature of singularities and how to resolve them, even in more complex varieties.
Toric Resolutions and Singularities
Definition and Role of Toric Resolutions
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Top images from around the web for Definition and Role of Toric Resolutions
Toric Spines at a Site of Learning | eNeuro View original
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- Toric resolutions are birational morphisms from smooth toric varieties to singular toric varieties that resolve singularities
- Constructed by subdividing the fan of the to obtain a smooth fan, corresponding to a
- Resolving singularities using toric methods involves modifying the combinatorial data of the fan while preserving the toric structure
- Allows for systematic study and resolution of singularities in toric varieties using of fans
Properties and Significance of Toric Resolutions
- Provide a powerful tool for understanding and resolving singularities in algebraic geometry
- Toric varieties serve as important examples and building blocks in algebraic geometry (affine spaces, projective spaces)
- Toric resolutions offer a combinatorial approach to studying singularities
- Singularities are encoded in the fan structure of the
- Resolving singularities corresponds to subdividing the fan to achieve smoothness
- Toric methods have applications beyond toric varieties
- Techniques and insights from toric resolutions can be applied to more general algebraic varieties
- Toric resolutions provide a bridge between combinatorics and algebraic geometry
Constructing Toric Resolutions
Subdivision Process
- Begin with the fan of the singular toric variety
- Subdivide the fan by adding new rays and cones until the resulting fan is smooth
- Introduce new rays generated by primitive vectors in the lattice
- Refine the cones to ensure they are all smooth
- Different choices of subdivision may lead to different toric resolutions with distinct properties
- is not unique
Toric Resolution Map
- Determined by the inclusion of the original fan into the subdivided fan
- Induces a between the corresponding toric varieties
- Maps the smooth toric variety onto the singular toric variety
- Resolves the singularities by "" the singular points
- map preserves the toric structure
- Compatibility with the torus action
- Allows for the study of the resolution using toric methods
Analyzing Toric Resolutions
Exceptional Locus
- The preimage of the singular locus under the toric resolution map
- Determined by studying the newly added rays and cones in the subdivided fan
- corresponds to the new rays and cones introduced during subdivision
- Geometry of the exceptional locus is encoded in the combinatorial structure of the subdivided fan
- Irreducible components and their intersections can be understood through the fan structure
- Provides insights into the structure of the resolved singularities
Intersection Theory and Discrepancies
- of exceptional divisors can be computed using linear relations among rays in the subdivided fan
- on the smooth toric variety can be studied combinatorially
- of exceptional divisors measure the difference between canonical divisors of smooth and singular toric varieties
- Determined by the combinatorial data of the fans
- Important invariants in the study of singularities and birational geometry
- Toric methods provide a computational framework for intersection theory and discrepancy calculations
- Allows for explicit computations and understanding of these invariants in the toric setting
Classifying Singularities for Toric Resolution
Types of Singularities
- Toric methods can resolve a wide range of singularities
- Quotient singularities arise from the action of a finite abelian group on a smooth toric variety
- Resolved by subdividing the fan according to the group action
- Gorenstein singularities are characterized by the existence of a Cartier divisor that is a multiple of the canonical divisor
- Toric resolutions are related to the combinatorics of the Gorenstein cone
- Terminal singularities are the mildest type of singularities in the minimal model program
- Toric resolutions can be studied using properties of the canonical divisor and singularity index
- Quotient singularities arise from the action of a finite abelian group on a smooth toric variety
Active Research Areas
- Classification of singularities that admit toric resolutions is an active area of research
- New classes of singularities, such as complexity-one T-varieties, are being investigated using toric methods
- Toric resolutions provide a framework for studying and classifying singularities
- Combinatorial approach allows for explicit constructions and computations
- Connections to other areas of algebraic geometry and commutative algebra
- Further developments in toric methods may lead to new insights and resolutions of singularities
- Potential for applications in birational geometry, mirror symmetry, and other areas of mathematics
Key Terms to Review (25)
Birational Morphism: A birational morphism is a type of morphism between algebraic varieties that establishes a rational equivalence between them, meaning they are 'almost' the same except for lower-dimensional subvarieties. This concept is crucial when studying how varieties can be transformed or simplified while maintaining essential geometric properties, especially in relation to singularities and resolutions. Understanding birational morphisms helps in the analysis of how complex geometric structures relate to simpler ones through techniques like toric resolutions.
Blowing Up: Blowing up is a process in algebraic geometry where a point or a subvariety of a scheme is replaced with a projective space or a more complex structure to resolve singularities. This technique helps transform spaces with undesirable features into smoother varieties, often allowing for a better understanding of their geometric properties and facilitating further analysis of their structure.
Chern classes: Chern classes are topological invariants associated with vector bundles that provide crucial information about the geometry and topology of manifolds. They help in understanding how vector bundles can be classified and relate to various cohomological properties, making them integral in areas like intersection theory and algebraic geometry.
Cohen-Macaulay: Cohen-Macaulay refers to a class of rings and their associated varieties that exhibit particularly nice properties in commutative algebra and algebraic geometry. Specifically, a ring is Cohen-Macaulay if the depth of the ring equals its Krull dimension, indicating that it has a well-behaved structure. This concept plays an important role in understanding the singularities of varieties, especially in the context of toric resolutions where the geometric properties are closely tied to algebraic characteristics.
Combinatorial Properties: Combinatorial properties refer to characteristics or features that can be understood and analyzed using combinatorial techniques, such as counting, arrangements, and the study of configurations in discrete structures. These properties play a crucial role in understanding how geometric objects behave, especially when resolving singularities in algebraic varieties and exploring toric varieties.
David Cox: David Cox is a prominent mathematician known for his contributions to algebraic geometry, particularly in the study of toric varieties and their resolutions. His work has provided important insights into the relationships between geometry, combinatorics, and algebra, especially in the context of resolving singularities through toric methods.
Discrepancies: In algebraic geometry, discrepancies are numerical invariants associated with a variety that help classify its singularities, specifically relating to how the singularity differs from a smooth variety. They play a crucial role in the Minimal Model Program (MMP) and are essential for understanding the nature of singularities, particularly when examining canonical and terminal singularities. The notion of discrepancies serves as a bridge to various resolutions and provides insight into the geometric structure of varieties.
Exceptional Locus: The exceptional locus is a set of points that arises during the process of a resolution of singularities, typically in algebraic geometry. It represents the points in the variety where the resolution introduces new divisors, often related to the complexities of singular points being resolved. Understanding this concept is essential for analyzing how singularities behave under various transformations and how resolutions can simplify the structure of a given space.
Fan structure: A fan structure is a combinatorial object that consists of a collection of cones in a vector space, organized in such a way that they fit together to form a fan. This concept is crucial for toric geometry, where it helps in studying varieties by connecting algebraic properties with geometric aspects, particularly when analyzing resolutions of singularities and the interaction between different algebraic objects.
Gorenstein Singularity: A Gorenstein singularity is a type of isolated singularity of a variety that possesses a dualizing sheaf, which allows for a well-defined canonical class. This property indicates that the local ring of the variety has nice homological properties, making it particularly interesting in algebraic geometry. Gorenstein singularities are often studied because they serve as a bridge between smooth varieties and more complicated singularities, and they can have rich geometric and algebraic structures.
Intersection Numbers: Intersection numbers are a fundamental concept in algebraic geometry that quantify how two subvarieties intersect within a larger variety. They provide a numerical representation of the intersection's complexity, capturing not just the number of intersection points but also their multiplicity and other geometric properties. Understanding intersection numbers is crucial for connecting various mathematical principles, such as duality, the behavior of maps between varieties, resolutions of singularities, and combinatorial aspects of geometrical structures.
Intersection Theory: Intersection theory is a branch of algebraic geometry that studies the intersections of algebraic varieties, providing a framework to understand their dimensions, multiplicities, and geometric properties. This concept is crucial for linking algebraic and geometric aspects of varieties, enabling the exploration of their relationships through tools like divisors and cohomology.
Polyhedral Cone: A polyhedral cone is a geometric object that consists of a finite set of vectors in a vector space, where all non-negative linear combinations of these vectors form the cone. This structure is essential in understanding toric varieties and the combinatorial data associated with fans, as it allows for the description of how various geometric objects are formed from these linear combinations.
Quotient Singularity: A quotient singularity is a type of singularity that arises when considering the quotient of a smooth variety by the action of a finite group. This type of singularity can often be resolved by toric methods, which involve constructing a new variety that 'smooths out' the singular point, making it easier to analyze the geometry and topology of the original space.
Rational Singularity: A rational singularity is a type of singular point on a variety where the local ring at that point behaves like a rational number in terms of its cohomological properties. This means that the singularity can be resolved in such a way that the resulting resolution retains certain desirable features, making it manageable in the study of algebraic varieties and their geometry. Rational singularities are particularly interesting because they allow for specific techniques like toric resolutions to be employed, which can simplify the analysis of singularities.
Self-intersection numbers: Self-intersection numbers are integers that represent the intersection of a geometric object with itself, particularly in the context of algebraic varieties and their embeddings. They provide valuable information about the geometry and singularities of these varieties, indicating how a divisor or a curve interacts with itself. Understanding self-intersection numbers is crucial when dealing with toric resolutions, as they help classify singularities and understand how resolutions modify the geometry.
Sheldon Katz: Sheldon Katz is a prominent mathematician known for his contributions to the field of algebraic geometry, particularly in the area of toric geometry and resolutions of singularities. His work often focuses on understanding how geometric structures can be studied through combinatorial methods, which is crucial in analyzing singularities and their resolutions.
Singular toric variety: A singular toric variety is a type of algebraic variety that arises from a fan in a lattice, where some of the cones in the fan correspond to points in the variety that are not smooth, meaning they have singularities. These varieties can be understood through combinatorial data and are important for studying their geometric properties, especially in relation to resolutions of singularities and their impact on the structure of the variety.
Smooth toric variety: A smooth toric variety is a type of algebraic variety that is defined by a fan, which consists of strongly convex polyhedral cones. These varieties are characterized by their nice geometric and algebraic properties, particularly the absence of singularities. Smooth toric varieties serve as an important tool in studying resolutions of singularities and understanding how complex algebraic structures can be simplified.
Subdivision Process: The subdivision process is a method in algebraic geometry that refines the structure of a variety by breaking it down into simpler, more manageable pieces. This approach is particularly important for studying toric varieties and their singularities, allowing mathematicians to systematically resolve complex geometric structures into simpler components, making analysis and computations more feasible.
Support Function: The support function is a mathematical tool used to describe and analyze convex sets in terms of their interaction with linear functionals. It essentially provides a way to express the maximum value of a linear function over a convex set, which is vital for understanding properties of polytopes, their duals, and structures in toric geometry. This concept helps bridge the gap between geometric objects and algebraic formulations, linking ideas from convex analysis to the study of algebraic varieties and their resolutions.
Terminal Singularity: A terminal singularity is a type of singularity in algebraic geometry where the local ring has a particular structure that implies it cannot be 'resolved' in a certain sense. This means that the singular point cannot be transformed into a smoother point through a sequence of blow-ups without introducing new singularities. Terminal singularities are important for understanding the minimal model program, as they represent one of the conditions that help categorize varieties based on their singular structures.
Toric Ideal: A toric ideal is an ideal in a polynomial ring that is generated by binomials corresponding to a combinatorial object called a fan, which encodes information about how to construct toric varieties. This concept links algebraic geometry and combinatorial geometry, revealing how properties of algebraic varieties can be understood through the lens of combinatorial data. Toric ideals play a crucial role in the study of toric morphisms and resolutions of singularities.
Toric Resolution: A toric resolution is a specific type of resolution of singularities that utilizes combinatorial and geometric properties of toric varieties to address singular points in algebraic varieties. This approach allows for a more structured analysis of the singularities by translating problems into the language of polyhedra and fans, making it easier to construct resolutions and study their properties.
Toric Variety: A toric variety is a special type of algebraic variety that is constructed from combinatorial data associated with convex polyhedra, particularly fans. They are particularly useful in algebraic geometry because they provide a way to study geometric objects through combinatorial methods, revealing connections between algebra, geometry, and topology.