6.4 Canonical and terminal singularities

Canonical and terminal singularities 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 Minimal Model Program, which aims to find simpler representations of complex geometric objects.

These singularities are defined by their discrepancies, which measure how far a variety is from being smooth. Canonical singularities 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

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• Let $X$ be a normal variety and $f : Y \to X$ be a resolution of singularities
• The canonical divisor $K_Y$ on $Y$ can be written as $f^*K_X + \sum a_i E_i$, where:
  • $E_i$ are the exceptional divisors
  • $a_i$ are the discrepancies
• A singularity is called canonical if all discrepancies $a_i \geq 0$ for every resolution of singularities
• A singularity is called terminal if all discrepancies $a_i > 0$ for every resolution of singularities

Computing Discrepancies

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

Classifying Singularities

Smooth Points and Quotient Singularities

• Smooth points are always terminal singularities
  • The discrepancies are strictly positive for any resolution
  • Smooth varieties have no exceptional divisors in their resolutions
• Quotient singularities 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 orbifold points (cyclic quotient singularities) and simple surface singularities (Du Val 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 $-2$
• Higher-dimensional singularities can be more challenging to classify
  • Often requires the computation of discrepancies for specific resolutions
  • Toric varieties 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 nef canonical divisor
  • Varieties with canonical or terminal singularities are the natural generalizations of smooth varieties in birational geometry
• The existence of minimal models (varieties with nef canonical divisor) is expected for varieties with canonical or terminal singularities
  • The existence may fail for worse singularities (log canonical or log terminal singularities)
  • The Minimal Model Conjecture 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
  • Flips and divisorial contractions are key steps in the MMP
  • These operations modify the variety while preserving the type of singularities
• The canonical ring $R(X, K_X) = \bigoplus_{n \geq 0} H^0(X, nK_X)$ is finitely generated 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 pluricanonical systems

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 Cone Theorem describes the structure of the nef cone and the existence of extremal rays
  • 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 Mori fiber space
  • 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