Glossary

6.4 Canonical and terminal singularities

4 min read•august 14, 2024

Canonical and are the mildest types of singularities in algebraic geometry. They're crucial for understanding the structure of varieties and play a key role in the , which aims to find simpler representations of complex geometric objects.

These singularities are defined by their , which measure how far a variety is from being smooth. have non-negative discrepancies, while terminal ones have strictly positive discrepancies. This classification helps simplify the study of varieties with singularities.

Canonical vs Terminal Singularities

Defining Canonical and Terminal Singularities

Top images from around the web for Defining Canonical and Terminal Singularities

1 of 1

Top images from around the web for Defining Canonical and Terminal Singularities

1 of 1
  • Let XX be a normal variety and f:Y→Xf : Y \to X be a
  • The KYK_Y on YY can be written as f∗KX+∑aiEif^*K_X + \sum a_i E_i, where:
    • EiE_i are the
    • aia_i are the discrepancies
  • A singularity is called canonical if all discrepancies ai≥0a_i \geq 0 for every resolution of singularities
  • A singularity is called terminal if all discrepancies ai>0a_i > 0 for every resolution of singularities

Computing Discrepancies

  • The discrepancy can be computed using the : (KY+Ei)⋅Ei=2pa(Ei)−2(K_Y + E_i) \cdot E_i = 2p_a(E_i) - 2
    • pa(Ei)p_a(E_i) is the arithmetic genus of the exceptional divisor EiE_i
  • The definition of canonical and terminal singularities is independent of the choice of resolution
    • This allows for a well-defined
    • Different resolutions will yield the same discrepancies for a given singularity

Classifying Singularities

Smooth Points and Quotient Singularities

  • are always terminal singularities
    • The discrepancies are strictly positive for any resolution
    • Smooth varieties have no exceptional divisors in their resolutions
  • arising from finite group actions on smooth varieties can be classified as canonical or terminal
    • The classification depends on the group action and the discrepancies of the exceptional divisors in the resolution
    • Examples of quotient singularities include (cyclic quotient singularities) and simple surface singularities ()

Surface and Higher-Dimensional Singularities

  • Surface singularities can be classified using the minimal resolution and the self-intersection numbers of the exceptional curves
    • Du Val singularities (ADE singularities) are canonical singularities
    • The minimal resolution of a Du Val singularity has exceptional curves with self-intersection numbers −2-2
  • Higher-dimensional singularities can be more challenging to classify
    • Often requires the computation of discrepancies for specific resolutions
    • provide a rich source of examples of canonical and terminal singularities in higher dimensions
    • The classification of higher-dimensional singularities is an active area of research

Significance of Singularities in Birational Geometry

Minimal Model Program (MMP)

  • Canonical and terminal singularities are the mildest types of singularities in the MMP
    • The MMP aims to find a birational model with mild singularities and
    • Varieties with canonical or terminal singularities are the natural generalizations of smooth varieties in
  • The existence of (varieties with nef canonical divisor) is expected for varieties with canonical or terminal singularities
    • The existence may fail for worse singularities (log canonical or )
    • The predicts the existence of minimal models for varieties with canonical or terminal singularities

Birational Invariance and Finite Generation

  • Canonical and terminal singularities are preserved under certain birational operations
    • and are key steps in the MMP
    • These operations modify the variety while preserving the type of singularities
  • The R(X,KX)=⨁n≥0H0(X,nKX)R(X, K_X) = \bigoplus_{n \geq 0} H^0(X, nK_X) is for varieties with canonical or terminal singularities
    • Finite generation is a crucial property in the study of birational geometry
    • It allows for the construction of canonical models and the study of the

Minimal Models for Varieties with Singularities

Minimal Model Program Techniques

  • The proof of the existence of minimal models relies on the techniques of the MMP
    • The describes the structure of the nef cone and the existence of
    • Extremal rays can be contracted to obtain a birational model with milder singularities (divisorial contractions or flips)
  • The termination of the MMP is a key step in the proof
    • The MMP terminates after finitely many steps, leading to a minimal model or a
    • The termination of flips is a crucial and challenging step, established in dimension 3 by Mori and in dimension 4 by Shokurov

Surface Case and Higher Dimensions

  • For surfaces, the existence of minimal models for varieties with canonical or terminal singularities follows from:
    • The classification of surface singularities
    • The fact that the MMP terminates for surfaces
  • In higher dimensions, the proof is more involved
    • Requires the Cone Theorem, the existence of divisorial contractions and flips, and the termination of the MMP
    • The existence of minimal models in arbitrary dimension for varieties with canonical or terminal singularities is still an open problem (the Minimal Model Conjecture)
    • The proof has been established in dimensions 3 and 4, but remains open in higher dimensions

Key Terms to Review (29)

Adjunction Formula: The adjunction formula is a crucial result in algebraic geometry that relates the canonical sheaf of a variety and its subvarieties. Specifically, it provides a way to compute the canonical divisor of a subvariety in terms of the ambient variety, allowing for insights into singularities and minimal models. This formula is particularly important for understanding how the geometry of subvarieties interacts with the properties of their parent varieties.
Birational Geometry: Birational geometry is a branch of algebraic geometry that studies the relationships between algebraic varieties through birational maps, which are isomorphisms defined on dense open subsets of the varieties. This approach allows mathematicians to classify varieties by examining their rational points and their singularities, leading to insights about their geometric properties and transformations. It plays a key role in understanding canonical and terminal singularities, as well as applications in Hilbert schemes and quotient schemes.
Birational Invariance: Birational invariance refers to a property of certain geometric objects, like varieties, that remains unchanged under birational transformations. In the context of algebraic geometry, this concept is crucial as it allows mathematicians to study and classify varieties up to birational equivalence, which essentially means that two varieties can be related through rational maps that are inverses on a dense open subset. This notion is particularly significant when discussing singularities, especially canonical and terminal singularities, as it helps in understanding how these properties behave under birational changes.
Canonical divisor: A canonical divisor is a divisor associated with a variety that encodes important geometric and topological information about the variety itself. It is often denoted by $K_X$ and can be viewed as a generalization of the notion of canonical forms in algebraic geometry, reflecting the duality between divisors and differentials. Understanding canonical divisors plays a crucial role in analyzing singularities, vanishing theorems, and applying the Riemann-Roch theorem to curves and surfaces.
Canonical Ring: The canonical ring of a projective variety is an algebraic object that captures information about its canonical divisor and serves as a tool to study the geometry of the variety. This ring, formed from sections of line bundles associated with the canonical divisor, plays a crucial role in understanding the singularities and properties of varieties, particularly in the classification of singularities like canonical and terminal singularities.
Canonical singularities: Canonical singularities are a type of singularity in algebraic geometry that behaves well under certain mathematical operations, particularly in the context of resolution of singularities. They are significant because they allow for the classification of varieties based on their singular points and play a crucial role in the study of minimal models and the Minimal Model Program. Canonical singularities appear when the canonical divisor has non-negative discrepancies, meaning that these singularities are considered to have mild behavior compared to other types of singularities.
Classification of singularities: The classification of singularities involves categorizing singular points of algebraic varieties based on their geometric and algebraic properties. This concept helps to understand the behavior of varieties near these points, revealing their structure and potential resolutions. By analyzing singularities, one can distinguish between different types such as canonical and terminal singularities, which play a crucial role in the minimal model program and birational geometry.
Cone Theorem: The Cone Theorem states that certain types of singularities in algebraic varieties can be classified based on the geometry of their associated cones. This theorem plays a crucial role in understanding canonical and terminal singularities, revealing how the local structure of a variety around a singular point relates to its global properties. It provides insights into how these singularities behave under various geometric transformations, helping to classify the types of singularities present.
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.
Divisorial Contractions: Divisorial contractions are morphisms that contract certain divisors on a variety, typically allowing one to simplify the structure of a space by collapsing the image of these divisors. This concept is crucial in the classification of singularities, particularly when identifying canonical and terminal singularities, which involve understanding how these contractions affect the properties of varieties and their singular points.
Du Val singularities: Du Val singularities are a class of surface singularities characterized by their specific algebraic structures, which are classified according to their types based on the associated Dynkin diagrams. These singularities are particularly important in the study of two-dimensional algebraic varieties, where they help in understanding the geometry and topology of surfaces, especially in the context of minimal models and resolutions.
Exceptional Divisors: Exceptional divisors are specific types of divisors that arise in the context of birational geometry, particularly during the resolution of singularities. They play a crucial role in understanding the behavior of the canonical class and the classification of singularities, such as canonical and terminal singularities. Exceptional divisors typically correspond to the components that are introduced when blowing up a variety at a point, and they help manage the discrepancies in the resolution process.
Extremal Rays: Extremal rays are specific directions in a convex cone associated with a given variety or space that represent the extreme behavior of divisors in the context of algebraic geometry. These rays help in understanding the properties of a variety, particularly in the study of its singularities and canonical models. They are crucial for determining the structure of the moduli space and understanding the interactions between different types of singularities.
Finitely generated: Finitely generated refers to a property of an algebraic object, typically a module or an algebra, where it can be constructed from a finite set of generators. This concept is essential in various areas of mathematics, including algebraic geometry, as it relates to the structure and behavior of algebraic varieties and their singularities. Understanding whether an algebraic object is finitely generated can provide insights into its dimensionality and complexity, which are critical when analyzing canonical and terminal singularities.
Flips: Flips are a type of birational transformation in algebraic geometry that modifies the structure of a variety to achieve a more desirable geometric property. Typically, this process involves taking a certain class of divisors and altering them to create a new variety, often resulting in a smoother or simpler singularity. Flips play a crucial role in the minimal model program, allowing mathematicians to transition between different models of algebraic varieties, particularly when dealing with canonical or terminal singularities.
Log canonical singularities: Log canonical singularities are a type of singularity in algebraic geometry characterized by their behavior under a log resolution. These singularities are important in the minimal model program and contribute to understanding the structure of varieties. They arise when the discrepancies associated with a resolution of singularities are at most zero, leading to the classification of singularities based on their geometric properties and their role in birational geometry.
Log Terminal Singularities: Log terminal singularities are a special class of singularities in algebraic geometry that arise when considering the minimal model program. They are defined as varieties with discrepancies that are non-negative when adjusted for log canonical thresholds, making them a milder form of singularities. Understanding log terminal singularities is crucial for the classification of algebraic varieties and plays a significant role in the study of their geometric properties.
Minimal Model Conjecture: The Minimal Model Conjecture is a fundamental hypothesis in algebraic geometry that posits the existence of a minimal model for any given algebraic variety, specifically in the context of varieties with certain singularities. This conjecture aims to simplify the study of varieties by asserting that there exists a 'canonical' form that retains essential geometric properties while eliminating extraneous complexities, particularly focusing on varieties with terminal and canonical singularities.
Minimal Model Program: The Minimal Model Program is a significant framework in algebraic geometry that aims to classify algebraic varieties by finding a 'minimal model' of them, which simplifies their structure. This program is closely linked to the study of singularities, particularly canonical and terminal singularities, as it provides tools to analyze these singularities and determine when they can be resolved or replaced with simpler models.
Minimal Models: Minimal models are algebraic varieties that are regarded as the simplest representatives of their respective birational equivalence classes, typically characterized by having the least complexity in terms of singularities and structure. These models help in understanding the geometry of varieties by allowing us to focus on essential features while simplifying or resolving complexities associated with non-minimal varieties. The concept is particularly relevant in the study of singularities, especially canonical and terminal singularities, as minimal models provide a foundation for classification and comparison.
Mori Fiber Space: A Mori fiber space is a specific type of algebraic variety that arises in the context of the Minimal Model Program (MMP), particularly when dealing with varieties that possess canonical or terminal singularities. These spaces are used to study the structure of varieties and the behavior of their singularities, providing insights into their birational properties and how they can be modified or simplified.
Nef Canonical Divisor: A nef canonical divisor is a divisor that is numerically effective, meaning it does not intersect negatively with any curves in the variety. It plays a crucial role in the classification of varieties and in understanding their geometric properties. In particular, a nef canonical divisor can help determine whether a variety has certain desirable features, such as being Fano or Kähler, and it is closely connected to the study of singularities, particularly canonical and terminal ones.
Orbifold Points: Orbifold points are special types of singular points on a space that exhibit a certain level of symmetry and allow for a local description similar to that of a quotient space. They generalize the notion of singularities, encompassing cases where the local structure can be understood through group actions, often arising in the study of algebraic varieties and their resolutions.
Pluricanonical Systems: Pluricanonical systems are collections of divisors associated with the multiples of the canonical divisor on a projective variety, which provide a way to study its geometry through sections of these divisors. These systems help in understanding the geometry of varieties with singularities, particularly focusing on the nature and properties of canonical and pluricanonical maps. By analyzing pluricanonical forms, one can derive significant information about the structure and classification of algebraic varieties, especially in the presence of singularities.
Quotient Singularities: Quotient singularities are specific types of singular points on algebraic varieties that arise from the action of a finite group on a smooth variety. These singularities are characterized by their local structure, which resembles the quotient of a smooth space by a group action, resulting in an isolated singularity that often has desirable properties in terms of resolutions and minimal models.
Resolution of singularities: Resolution of singularities is a process in algebraic geometry that aims to replace a singular variety with a new variety that has no singularities. This is crucial for understanding the geometry and topology of spaces, as well as for simplifying calculations. By resolving singularities, we can gain insights into the behavior of functions near these problematic points and study the structure of varieties more effectively.
Smooth points: Smooth points on a variety are points where the local geometry behaves nicely, meaning that the tangent space is well-defined and has the expected dimension. These points do not exhibit any singular behavior, which contrasts with singular points where the geometry can be more complicated and irregular. In the context of canonical and terminal singularities, identifying smooth points is crucial as they often play a key role in understanding the structure and classification of singularities.
Terminal Singularities: Terminal singularities are a type of singular point on an algebraic variety where the resolution of singularities results in a variety that has a non-negative canonical divisor. This concept is crucial in the study of minimal models, as terminal singularities are considered 'mild' types of singularities, making them manageable in geometric terms. Understanding terminal singularities helps classify varieties based on their singularity types and informs the study of birational geometry.
Toric Varieties: Toric varieties are a special class of algebraic varieties that are constructed from combinatorial data, specifically from fans or polyhedral cones. They provide a bridge between algebraic geometry and combinatorial geometry, allowing for a geometric interpretation of algebraic objects. Their structure is deeply tied to torus actions, which means they can be studied through the lens of algebraic groups and their actions on varieties.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide