🌿Algebraic Geometry Unit 4 – Divisors and Line Bundles

Key Concepts and Definitions

• Divisors represent codimension-1 subvarieties or formal sums of subvarieties on an algebraic variety or scheme
• Line bundles are locally free sheaves of rank 1 on an algebraic variety or scheme
• Picard group $\text{Pic}(X)$ classifies isomorphism classes of line bundles on a variety $X$
• Chow group $A^1(X)$ is the group of divisors modulo linear equivalence on a variety $X$
• Canonical divisor $K_X$ associated to the canonical bundle $\omega_X$ on a smooth variety $X$
• Cartier divisors are locally principal divisors defined by a single equation
• Weil divisors are formal sums of codimension-1 subvarieties with integer coefficients
• Linear equivalence relation between divisors if their difference is a principal divisor

Divisors: Types and Properties

• Effective divisors have non-negative coefficients in their formal sum representation
• Principal divisors are divisors of rational functions on a variety
• Ample divisors have a positive multiple that is very ample (defines an embedding into projective space)
• Nef divisors intersect non-negatively with every curve on the variety
• Divisors on curves correspond to formal sums of points with integer coefficients
  • Degree of a divisor on a curve is the sum of its coefficients
• Divisors on surfaces can be represented by formal sums of curves
• Divisors on higher-dimensional varieties involve codimension-1 subvarieties
• Intersection product of divisors extends the intersection theory of subvarieties

Line Bundles: Basics and Construction

• Line bundles are geometric objects that generalize the notion of a vector bundle of rank 1
• Sections of a line bundle are global functions that locally generate the line bundle
• Transition functions of a line bundle are invertible regular functions on the overlaps of an open cover
• First Chern class $c_1(L)$ of a line bundle $L$ is a cohomological invariant measuring its "twistedness"
• Tensor product of line bundles corresponds to the sum of divisors
• Dual line bundle corresponds to the negative of a divisor
• Pullback of a line bundle under a morphism of varieties
• Constructing line bundles using Čech cocycles or divisors
  • Čech cocycles are transition functions satisfying the cocycle condition
  • Divisors define line bundles via the sheaf of rational functions with prescribed poles and zeros

Relationship Between Divisors and Line Bundles

• Divisor class group $\text{Cl}(X)$ is isomorphic to the Picard group $\text{Pic}(X)$ for smooth varieties
• Cartier divisors correspond to line bundles, while Weil divisors may not
• Effective divisors correspond to global sections of the associated line bundle
• Canonical divisor corresponds to the canonical bundle (top exterior power of the cotangent bundle)
• Degree of a divisor on a curve equals the degree of the corresponding line bundle
• Intersection product of divisors relates to the tensor product of line bundles
• Chern class of a line bundle determines the associated divisor up to linear equivalence
• Riemann-Roch theorem relates the dimension of global sections of a line bundle to its degree and the canonical divisor

Intersection Theory and Chern Classes

• Intersection theory studies the intersection of subvarieties and the resulting multiplicities
• Chern classes are cohomological invariants associated to vector bundles, including line bundles
• First Chern class $c_1(L)$ of a line bundle $L$ is an element of the second cohomology group $H^2(X, \mathbb{Z})$
• Chern classes satisfy certain axioms, such as functoriality under pullbacks and the Whitney product formula
• Intersection product of divisors can be computed using Chern classes and the cup product in cohomology
• Adjunction formula relates the canonical divisor of a subvariety to the restriction of the canonical divisor and the normal bundle
• Riemann-Roch theorem for surfaces involves the first and second Chern classes of the tangent bundle
• Hirzebruch-Riemann-Roch theorem generalizes the Riemann-Roch theorem to higher dimensions using Chern characters

Applications in Algebraic Geometry

• Classification of algebraic curves using divisors and the degree-genus formula
• Studying the geometry of algebraic surfaces using the intersection theory of curves
• Determining the existence of rational points on varieties using the Brauer-Manin obstruction
• Computing the Picard group and the Neron-Severi group of a variety
• Studying the moduli space of line bundles on a variety (Picard scheme)
• Investigating the geometry of projective embeddings using very ample divisors
• Applying the Riemann-Roch theorem to compute dimensions of linear systems
• Classifying algebraic varieties using numerical invariants derived from divisors and Chern classes (Kodaira dimension, Chern numbers)

Computational Techniques and Examples

• Computing Gröbner bases to determine the ideal of a subvariety and its associated divisor
• Using toric geometry to study divisors and line bundles on toric varieties
• Calculating the intersection matrix of curves on a surface
• Determining the Picard group of a variety using cohomological techniques (Čech cohomology, sheaf cohomology)
• Applying the Riemann-Roch theorem to compute the dimension of the space of global sections of a line bundle
  • Example: Computing the dimension of the complete linear system associated to a divisor on a curve
• Using the adjunction formula to determine the canonical divisor of a subvariety
• Computing Chern classes of vector bundles using the splitting principle and the Whitney product formula
• Utilizing software packages like Macaulay2 or Sage to perform computations in algebraic geometry

Advanced Topics and Open Problems

• Moduli spaces of divisors and line bundles (Picard schemes, Jacobians of curves)
• Néron-Severi group and the cone of ample divisors
• Hodge conjecture relating the cohomology of a variety to the intersection of algebraic cycles
• Minimal model program and the role of canonical divisors in birational geometry
• Donaldson-Thomas invariants and their relation to Chern classes and intersection theory
• Arakelov geometry and the study of divisors and line bundles on arithmetic varieties
• Tropical geometry and its connections to divisor theory and intersection theory
• Derived categories and the role of line bundles in the study of derived equivalences and Fourier-Mukai transforms