Cartier classes represent a fundamental concept in algebraic geometry, specifically within the study of divisors on schemes. They arise from Cartier divisors, which are more refined than Weil divisors, allowing for a precise treatment of line bundles and cohomology. Understanding Cartier classes is essential for advanced topics like intersection theory and moduli spaces.
What Is a Cartier Divisor?
A Cartier divisor on a scheme X is defined locally by a collection of rational functions that agree on overlaps. Formally, it is represented by an open cover {U_i} of X and invertible sheaves O_{U_i} together with sections f_i in K(X)(U_i), the function field, such that f_i / f_j is a regular function on U_i ∩ U_j.
Two such representations are equivalent if they differ by units in the structure sheaf. This local principal nature makes Cartier divisors ideal for schemes with mild singularities.
How Do Cartier Classes Form from Cartier Divisors?
Cartier classes are equivalence classes of Cartier divisors under linear equivalence. Just as principal divisors are those coming from global rational functions, two Cartier divisors D and D’ are linearly equivalent if D – D’ is principal locally.
The group of Cartier classes, denoted CaCl(X) or sometimes Cl(X) for normal schemes, is the quotient of the group of Cartier divisors by principal ones. This group maps to the Picard group Pic(X), which classifies line bundles up to isomorphism.
Key Insight: On integral schemes, the map from Cartier classes to the class group (Weil classes) is an isomorphism under certain conditions, like normality.
What Is the Difference Between Cartier Classes and Weil Classes?
Weil divisors are formal sums of codimension-one prime divisors, while Cartier divisors require a principal description locally. Cartier classes are “rational” in nature, making them suitable for non-normal schemes where Weil divisors may not suffice.
For example, on a nodal curve, a Cartier class can distinguish components crossing the node, whereas Weil classes might not. This distinction is crucial in birational geometry.
Why Are Cartier Classes Important in Algebraic Geometry?
Cartier classes play a pivotal role in computing intersection numbers, Chern classes, and K-theory. They enable the definition of the Chow group of cycles modulo rational equivalence via refinements.
Simple Example: On projective space P^n, Cartier classes correspond to degrees of hypersurfaces, mirroring the numerical class group Z.
What Are Common Applications of Cartier Classes?
In moduli problems, Cartier classes classify determinantal line bundles. They also appear in the study of reflexive sheaves and minimal model programs, where reflexivity aligns with Cartier properties.
In summary, Cartier classes provide a flexible framework bridging divisors, line bundles, and higher cohomology, essential for modern algebraic geometry research.
People Also Ask
How do Cartier classes relate to the Picard group?
The Picard group Pic(X) is isomorphic to the group of Cartier classes on locally factorial schemes, as each class corresponds to a line bundle via associated sheaves.
Can Cartier classes be defined on singular varieties?
Yes, they extend naturally to singular schemes, unlike strict Weil divisors, making them versatile for stacks and quotients.
What is an example of a non-trivial Cartier class?
On the blow-up of P^2 at a point, the exceptional divisor forms a Cartier class of order 1, generating part of the Picard group Z ⊕ Z.
