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0 · very ample divisor
1 · relative cartier divisor worksheet
2 · pullback of divisor
3 · locally principal divisor
4 · effective cartier divisor
5 · cartier divisors pdf
6 · cartier divisors and linear systems
7 · cartier divisor worksheet pdf
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Learn how to define and manipulate Cartier divisors on schemes, which are pairs of rational sections of line bundles satisfying certain conditions. See the relation between Cartier divisors .Learn the definitions and properties of Weil and Cartier divisors on algebraic varieties, and how they are related to line bundles and linear systems. See examples of divisors on Pn, P2, and .
In algebraic geometry, divisors are a generalization of codimension -1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields.L. [div(s)] = ordV X (s)[V ]; V. where V ranges over codimension-one subvarieties of X. Intuitively, we think of [div(s)] as \zeros of s" \poles of s". If X is locally factorial, then every Weil divisor can be obtained as the divisor of. ome line bundle. Moreover, we can reconstruct Weil divisor (by a process I prefer not to go into at this time.On smooth varieties, Weil divisors are in bijection with Cartier divisors. On singular varieties, there may be Weil divisors that cannot be given as Cartier divisors, or non-trivial Cartier divisors for which the operation above produces a zero Weil divisor. Weil divisors naturally form an abelian group (we just add the linear combinations .Cartier divisors and invertible sheaves are equivalent (categorically). Given D 2 DivC(X), then we get an invertible subsheaf in K, locally it's fiO, the O-submodule generated by fi by construction it is locally isomorphic to O. Conversely if L K is locally isomorphic to O, A system of local generators de nes the data as above.
A Cartier divisor is called principal if it is in the image of ( X;K). Two Cartier divisors Dand D 0 are called linearly equivalent, denoted D˘D 0 , if and only if the di erence is principal.A Cartier divisor on X is a section of the sheaf K(X)/O× . Using the construction of principal divisors, we obtain a map from Cartier divisors to Weil divisors: if the Cartier divisor is represented on some open subset U of X by the rational function f ∈ K(X), then the Weil divisorAn effective Cartier divisor on $S$ is a closed subscheme $D \subset S$ whose ideal sheaf $\mathcal{I}_ D \subset \mathcal{O}_ S$ is an invertible $\mathcal{O}_ S$-module. Thus an effective Cartier divisor is a locally principal closed subscheme, but .
the Cartier divisors are isomorphic to the subgroup of locally principal Weil divisors, as claimed at the beginning of the section. So, on normal schemes (where Weil divisors can be defined), the Cartier divisors are a subset of the Weil divisors. If our scheme is not regular or not locally factorial, they do not have to be the same. Example 1.4.A Cartier divisor is called principal if it is in the image of ( X;K). Two Cartier divisors Dand D 0 are called linearly equivalent, denoted D˘D 0 if and only if the di erence is principal. On a scheme X X, a Cartier divisor is a global section of the sheaf K ∗/O ∗ 𝒦 * / 𝒪 *, where K ∗ 𝒦 * is the multiplicative sheaf of meromorphic functions, and O ∗ 𝒪 * the multiplicative sheaf of invertible regular functions (the units of the structure sheaf).
very ample divisor
In algebraic geometry, divisors are a generalization of codimension -1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields.L. [div(s)] = ordV X (s)[V ]; V. where V ranges over codimension-one subvarieties of X. Intuitively, we think of [div(s)] as \zeros of s" \poles of s". If X is locally factorial, then every Weil divisor can be obtained as the divisor of. ome line bundle. Moreover, we can reconstruct Weil divisor (by a process I prefer not to go into at this time.
On smooth varieties, Weil divisors are in bijection with Cartier divisors. On singular varieties, there may be Weil divisors that cannot be given as Cartier divisors, or non-trivial Cartier divisors for which the operation above produces a zero Weil divisor. Weil divisors naturally form an abelian group (we just add the linear combinations .
Cartier divisors and invertible sheaves are equivalent (categorically). Given D 2 DivC(X), then we get an invertible subsheaf in K, locally it's fiO, the O-submodule generated by fi by construction it is locally isomorphic to O. Conversely if L K is locally isomorphic to O, A system of local generators de nes the data as above.
A Cartier divisor is called principal if it is in the image of ( X;K). Two Cartier divisors Dand D 0 are called linearly equivalent, denoted D˘D 0 , if and only if the di erence is principal.
A Cartier divisor on X is a section of the sheaf K(X)/O× . Using the construction of principal divisors, we obtain a map from Cartier divisors to Weil divisors: if the Cartier divisor is represented on some open subset U of X by the rational function f ∈ K(X), then the Weil divisorAn effective Cartier divisor on $S$ is a closed subscheme $D \subset S$ whose ideal sheaf $\mathcal{I}_ D \subset \mathcal{O}_ S$ is an invertible $\mathcal{O}_ S$-module. Thus an effective Cartier divisor is a locally principal closed subscheme, but .the Cartier divisors are isomorphic to the subgroup of locally principal Weil divisors, as claimed at the beginning of the section. So, on normal schemes (where Weil divisors can be defined), the Cartier divisors are a subset of the Weil divisors. If our scheme is not regular or not locally factorial, they do not have to be the same. Example 1.4.
A Cartier divisor is called principal if it is in the image of ( X;K). Two Cartier divisors Dand D 0 are called linearly equivalent, denoted D˘D 0 if and only if the di erence is principal.
relative cartier divisor worksheet
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pullback of divisor
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cartier divisor|effective cartier divisor
