Relative Picard Number

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tags: - colorclass/differential geometry ---The Relative Picard Number is an important concept in algebraic geometry, especially within the framework of the Minimal Model Program (MMP) and the study of Mori fiber spaces. It serves as a measure of the complexity of the divisor class group of a projective variety relative to a fibration or morphism to another variety. Understanding the relative Picard number is crucial for analyzing the structure of algebraic varieties and their birational properties.

Definition

Given a projective morphism $f:X\to Y$ between two algebraic varieties $X$ and $Y$, the relative Picard number, denoted $\rho (X/Y)$, is defined as the rank of the Néron-Severi group $NS(X/Y)$ of $X$ over $Y$. This group represents the group of divisors on $X$ that are numerically equivalent over $Y$, modulo numerical equivalence.

In simpler terms, it captures the dimension of the space of “geometrically distinct” ways one can map curves into $X$, taking into account the fibration over $Y$.

Importance in Algebraic Geometry

- Birational Geometry and MMP: The relative Picard number plays a pivotal role in the MMP, which seeks to classify algebraic varieties by simplifying their structure through birational transformations. The relative Picard number helps in understanding the complexity of these transformations, especially when dealing with fibrations in Mori fiber spaces, where it must be equal to one. This condition ensures that the fibration is “minimally complex” in a certain sense, making it easier to analyze the geometric structure of the variety.

- Mori Fiber Spaces: For Mori fiber spaces, a low relative Picard number (specifically, one) indicates a certain minimality or simplicity of the fibration structure. It implies that the fibration does not admit further non-trivial fibrations that could simplify the variety’s structure. This simplicity is essential for the decomposition and classification of varieties within the MMP.

- Measure of Fibration Complexity: The relative Picard number quantifies the complexity of the fibration $f:X\to Y$ by measuring how many “directions” of divisor classes exist that are not accounted for by the base $Y$. A higher relative Picard number indicates a richer structure of the fibered variety, with more degrees of freedom in the choice of divisors.

Calculation and Challenges

Calculating the relative Picard number can be challenging, as it often involves intricate geometric and topological analysis of the variety and its fibration. Techniques from Hodge theory, intersection theory, and the study of algebraic cycles are typically employed to tackle these calculations.

Applications

- Deformation Theory: Understanding the relative Picard number is also relevant in deformation theory, where it can provide insights into how the structure of a variety changes under small deformations, especially in the context of families of varieties parameterized by the base $Y$.

- Study of Algebraic Cycles: The investigation of algebraic cycles and their equivalence relations (like numerical and rational equivalence) is deeply connected to the study of the Picard groups and Néron-Severi groups, offering a window into the arithmetic and geometric properties of varieties.

The relative Picard number, by encapsulating the diversity and complexity of divisor classes in a fibration, offers a numerical glimpse into the geometric intricacies of algebraic varieties, enabling deeper understanding and classification of complex geometric structures.