4.2 Picard group and line bundles
4 min read•august 14, 2024
Line bundles and the are key concepts in algebraic geometry. They connect divisors, sheaves, and geometric properties of varieties. Understanding these ideas helps us grasp how algebraic structures relate to the geometry of varieties.
The Picard group classifies line bundles on a variety, while line bundles correspond to divisors. This relationship lets us study geometric properties using algebraic tools. It's crucial for understanding ample line bundles, embeddings, and cohomology of varieties.
Picard Group of Algebraic Varieties
Definition and Relationship to Divisors
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- The Picard group of an algebraic variety X, denoted Pic(X), is the group of isomorphism classes of line bundles on X under the operation
- Pic(X) is isomorphic to the group of divisors on X modulo linear equivalence
- A divisor D on X is a formal sum of codimension-1 subvarieties of X with integer coefficients
- The group of divisors on X is denoted Div(X)
- Two divisors D and D' are linearly equivalent if their difference is the divisor of a rational function on X
- This defines an equivalence relation on Div(X), and the quotient group Div(X)/~ is isomorphic to Pic(X)
Properties and Homomorphisms
- The of a divisor D is the sum of its coefficients
- The degree map Pic(X) → Z is a group homomorphism
- The Picard group captures important geometric information about the variety X
- Elements of Pic(X) correspond to different ways of twisting line bundles on X
- The Picard group is a contravariant functor from the category of algebraic varieties to the category of abelian groups
- A morphism f: X → Y induces a group homomorphism f*: Pic(Y) → Pic(X) by pulling back line bundles
Line Bundles Associated to Divisors
Construction and Properties
- Given a divisor D on an algebraic variety X, one can construct a line bundle L(D) on X
- The construction of L(D) involves defining a sheaf of sections locally using the coefficients of D and gluing these local descriptions together
- The line bundle L(D) has the property that its space of global sections H⁰(X, L(D)) is isomorphic to the space of rational functions f on X such that div(f) + D ≥ 0
- This means that the global sections of L(D) correspond to rational functions with poles and zeros prescribed by the divisor D
Operations and Duality
- The tensor product of line bundles corresponds to the addition of divisors: L(D) ⊗ L(D') ≅ L(D + D')
- The dual of a line bundle corresponds to the negative of a divisor: L(D)* ≅ L(-D)
- The inverse of a line bundle L in Pic(X) is given by its dual L*
- A line bundle L is ample if some positive tensor power of L defines an embedding of X into projective space
- Ample line bundles play a crucial role in the classification of algebraic varieties (Kodaira embedding theorem)
Picard Group Computation for Examples
Basic Examples
- The Picard group of projective space Pⁿ is isomorphic to Z, generated by the class of the hyperplane bundle O(1)
- The line bundles on Pⁿ are precisely the tensor powers O(d) for d ∈ Z
- The Picard group of a smooth curve C of genus g is isomorphic to the Jacobian variety of C, which is a g-dimensional abelian variety
- The degree 0 line bundles on C form a subgroup of Pic(C) isomorphic to the Jacobian
Product and Blow-up Formulas
- The Picard group of the product of two varieties X and Y is isomorphic to the direct sum of their Picard groups: Pic(X × Y) ≅ Pic(X) ⊕ Pic(Y)
- This follows from the fact that a line bundle on a product is determined by its restrictions to the factors
- The Picard group of a blow-up of a variety X at a point is isomorphic to the direct sum of Pic(X) and a copy of Z generated by the exceptional divisor
- The exceptional divisor is the preimage of the blown-up point under the blow-up map
- Computing the Picard group of a general algebraic variety can be a challenging problem and often involves advanced techniques from algebraic geometry and cohomology theory (Čech cohomology, exponential sequence)
Line Bundles vs Invertible Sheaves
Relationship and Equivalence
- An invertible sheaf on an algebraic variety X is a locally free sheaf of rank 1
- The global sections of an invertible sheaf form a line bundle on X, and conversely, every line bundle arises as the global sections of an invertible sheaf
- This establishes a bijective correspondence between line bundles and invertible sheaves on X
- The tensor product of invertible sheaves corresponds to the tensor product of line bundles
- The inverse of an invertible sheaf is given by its dual sheaf
Sheaf-Theoretic Perspective
- The group of isomorphism classes of invertible sheaves on X under the tensor product operation is isomorphic to the Picard group Pic(X)
- Invertible sheaves provide a sheaf-theoretic perspective on line bundles and are often used in the study of cohomology and the construction of moduli spaces of algebraic varieties
- The of an invertible sheaf encode important geometric information (Serre duality, )
- The use of invertible sheaves allows for the application of powerful tools from homological algebra and sheaf theory to the study of line bundles and the Picard group (derived categories, spectral sequences)
Key Terms to Review (16)
Alexander Grothendieck: Alexander Grothendieck was a French mathematician who made significant contributions to algebraic geometry, particularly in the development of schemes and sheaf theory. His revolutionary ideas transformed the field by introducing new concepts such as the étale topology, to study algebraic varieties, which have deep implications in intersection theory, vector bundles, and cohomology.
Ample Line Bundle: An ample line bundle is a line bundle that has the property that some positive tensor power of it can embed the variety into projective space. This means that an ample line bundle can be thought of as providing enough global sections to allow for a geometric realization of the variety in a higher-dimensional projective space, which connects to key ideas such as divisors, sheaves, and the geometry of varieties.
Cohomology Groups: Cohomology groups are algebraic structures that provide a way to classify and measure the global properties of topological spaces, particularly in relation to sheaves and their cohomology. They arise from the study of the relationship between local data (like sections of a sheaf) and global phenomena (like the topology of a space). This connection is especially important when discussing line bundles and their properties, as they help determine how these bundles behave under various transformations and mappings.
Degree: In algebraic geometry, the degree of a divisor or a curve is a numerical invariant that captures important information about the geometric properties of a variety. It can reflect how many points of intersection occur with a line or a plane, how many times a curve wraps around a point, and helps in understanding the overall shape and dimension of algebraic objects. Degree plays a key role in various theorems and concepts like duality and line bundles, linking the algebraic structure with geometric intuition.
Dual Line Bundle: A dual line bundle is a construction in algebraic geometry that assigns to each point of a variety a dual vector space, essentially representing the space of linear functions on the fibers of the original line bundle. This concept allows for a way to study properties of line bundles by considering their duals, creating a relationship between the original bundle and its dual that can reveal additional geometrical and topological information.
Émile Picard: Émile Picard was a prominent French mathematician known for his contributions to several areas of mathematics, including algebraic geometry, complex analysis, and the theory of functions. His work is particularly significant in the context of line bundles and the Picard group, which are essential concepts in understanding the classification of line bundles over algebraic varieties.
First Chern Class: The first Chern class is a topological invariant associated with complex line bundles, representing a way to measure the curvature of the bundle. It is an essential tool in algebraic geometry and differential geometry, providing important information about the geometry of the underlying space. This class helps connect line bundles to cohomology, making it crucial for understanding the Picard group and the classification of line bundles over a variety.
Grothendieck's Theorem on the Picard Group: Grothendieck's Theorem on the Picard Group states that the Picard group of a proper variety over an algebraically closed field is isomorphic to the group of line bundles modulo isomorphism. This theorem connects the study of line bundles, which are important in understanding divisors and line sheaves, to the more abstract structure of the Picard group, highlighting its role in classifying line bundles up to isomorphism.
Picard Group: The Picard group is a fundamental concept in algebraic geometry that classifies line bundles (or divisor classes) on a given algebraic variety. It captures important information about the geometry of the variety, including its rational functions and divisors, and allows for the study of how these structures interact under various transformations.
Picard Scheme: The Picard scheme is a geometric object that parametrizes line bundles on a given algebraic variety, offering insight into the structure of line bundles and their relationships. It provides a more nuanced understanding of the Picard group, which captures isomorphism classes of line bundles, and allows one to study families of line bundles over a base scheme. This construction plays a key role in various areas of algebraic geometry, particularly in the study of divisors and cohomology.
Projective variety: A projective variety is a type of geometric object that is defined as the zero set of homogeneous polynomials in projective space. These varieties can be thought of as the solutions to equations that describe geometric shapes, and they are important because they allow us to work with properties that are invariant under projective transformations, making them a central concept in algebraic geometry.
Riemann-Roch Theorem: The Riemann-Roch Theorem is a fundamental result in algebraic geometry that provides a way to compute the dimensions of space of meromorphic sections of line bundles on algebraic curves and varieties. This theorem links the geometry of curves to algebraic data associated with divisors, allowing for deeper insights into the properties of algebraic varieties and their functions.
Sheaf Cohomology: Sheaf cohomology is a powerful tool in algebraic geometry that studies the global sections of sheaves over a topological space, providing insights into the geometric and topological properties of varieties. It connects local properties of sheaves to their global behavior, making it essential for understanding various features like duality, line bundles, and moduli spaces.
Smooth variety: A smooth variety is a type of algebraic variety that has no singular points, meaning it behaves nicely in terms of its geometric and algebraic properties. Smoothness ensures that at every point in the variety, the local structure resembles that of an affine space, which is crucial for various mathematical concepts like intersection theory and cohomology. This concept plays a vital role in understanding line bundles and their classifications, as well as in advanced results like the Grothendieck-Riemann-Roch theorem.
Tensor Product: The tensor product is an operation that combines two vector spaces into a new vector space, which captures the relationships between the two spaces. It is crucial in various mathematical contexts, including the study of multilinear maps and modules. In the realm of algebraic geometry, the tensor product is particularly important for understanding line bundles and their properties, as it allows for the combination of sections from different line bundles into new ones.
Trivial Line Bundle: A trivial line bundle is a specific type of line bundle over a topological space that is globally isomorphic to the product of that space with the underlying field, typically represented as $$X \times \mathbb{C}$$ or $$X \times \mathbb{R}$$. It essentially means that the fibers over each point in the space are all the same and can be thought of as a constant bundle. This concept plays a crucial role in understanding the Picard group, as it helps define the notion of isomorphism and provides a base for classifying line bundles over a given space.