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

Let $X$ be a normal variety and $f : Y \to X$ be a resolution of singularities
The canonical divisor $K_Y$ $K_{Y}$ on $Y$ $Y$ can be written as $f^*K_X + \sum a_i E_i$ $f^{*} K_{X} + \sum a_{i} E_{i}$ , 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: $(K_Y + E_i) \cdot E_i = 2p_a(E_i) - 2$ $(K_{Y} + E_{i}) \cdot E_{i} = 2 p_{a} (E_{i}) - 2$
- $p_a(E_i)$ 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 $R(X, K_X) = \bigoplus_{n \geq 0} H^0(X, nK_X)$ $R (X, K_{X}) = ⨁_{n \geq 0} H^{0} (X, n K_{X})$ 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

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