Mori Fiber Spaces

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tags: - colorclass/differential geometry ---Mori fiber spaces are an essential concept within the Minimal Model Program (MMP), a comprehensive framework in algebraic geometry aimed at classifying higher-dimensional algebraic varieties. Named after Shigefumi Mori, who made significant contributions to the MMP and birational geometry, Mori fiber spaces play a pivotal role in understanding the structure of algebraic varieties, especially in cases where minimal models do not exist due to certain geometric or topological constraints.

Definition and Structure

A Mori fiber space is a specific type of fibration, $\phi :X\to B$, where $X$ is a projective variety, and $B$ is a lower-dimensional variety (base), satisfying the following conditions:

1. Fibers: The fibers of $\phi$ (the preimages of points in $B$) are connected varieties. The dimension of the fibers is positive, meaning that $\dim (X)>\dim (B)$.

2. Relative Picard Number: The relative Picard number of the fibration, which measures the rank of the group of line bundles on $X$ modulo numerical equivalence relative to the fibration, is equal to one. This condition ensures that the fibration is relatively minimal in terms of divisors.

3. Singularities: $X$ has terminal singularities, a particular type of mild singularity that allows for the application of the MMP. Additionally, the fibration is expected to satisfy certain positivity properties regarding the canonical divisor of $X$ relative to the base $B$.

Role in the Minimal Model Program

- Decomposition of Varieties: In the context of the MMP, Mori fiber spaces arise when attempting to simplify the structure of a given variety via birational transformations. If a variety cannot be directly simplified to a minimal model (often due to its canonical divisor being not nef, which informally means that it has “negative curvature”), the MMP aims to express it as a Mori fiber space. This represents the variety as a fibration over a simpler base, with the general fibers being simpler varieties themselves, often of lower dimension.

- Canonical Divisor Behavior: The construction of Mori fiber spaces is particularly relevant for varieties whose canonical divisors have negative intersection with certain curves, leading to the notion of varieties of “Fano type” or similar conditions. The behavior of the canonical divisor plays a crucial role in determining the structure and singularities of both the fibers and the total space.

Examples and Applications

- Fano Varieties: The fibers of Mori fiber spaces are often Fano varieties, which are varieties whose anticanonical divisors are ample, reflecting positive curvature properties.

- Classification and Structure: Mori fiber spaces contribute to the classification of algebraic varieties by providing a structured way to decompose varieties into components with understood geometry. This decomposition helps in mapping the vast landscape of algebraic varieties and understanding their geometric properties.

- Birational Geometry and Moduli Spaces: The study of Mori fiber spaces is crucial in birational geometry, impacting the understanding of moduli spaces, which classify varieties and geometric structures up to certain equivalences.

Mori fiber spaces exemplify the intricate relationship between geometry and topology in algebraic varieties, showcasing how complex varieties can be decomposed into more manageable pieces. This approach not only advances the classification of algebraic varieties but also deepens our understanding of their geometric and topological intricacies.