cartier divisor 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.
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1 · relative cartier divisor worksheet
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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 .
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 .
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Learn what Cartier divisors are and how they relate to Weil divisors, invertible sheaves and toric varieties. See examples of Cartier divisors on a quadric cone and a toric variety, and their .Learn about the definitions and properties of Weil and Cartier divisors, and how they relate to the Picard group of an irreducible variety. See examples, proofs, and applications of smoothness .
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 .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 locally principal closed subscheme of S is a closed subscheme whose sheaf of ideals is locally generated by a single element. An effective Cartier divisor on S is a closed subscheme D ⊂ S .
Cartier divisors. January 31, 2011. 1 Examples. Example 1.1. Let X be the affine quadric cone X = Spec k[X, Y, Z]/(XY −Z2). We will show CaCl = 0 and Cl = Z/2. Example 1.2. Let X be a double .
Cartier divisor on X, then it restricts to a closed subscheme on Y, locally cut out by one equation. If you are fortunate and this equation doesn’t vanish on any associated point of Y, then you get .31.18 Relative effective Cartier divisors. The following lemma shows that an effective Cartier divisor which is flat over the base is really a “family of effective Cartier divisors” over the base. For example the restriction to any fibre is an effective Cartier divisor.
31.14 Effective Cartier divisors and invertible sheaves Since an effective Cartier divisor has an invertible ideal sheaf (Definition 31.13.1 ) the following definition makes sense. Definition 31.14.1 .NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil’s book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil’s book, and the originality of these notes lies in the mistakes. I learned some of this .The point is that in a factorial domain, the height one prime ideals are principal. By definition a Weil divisor gives a height one prime ideal in the local ring a each point (this is the ideal that cuts out the Weil divisor), and if this local ring is factorial, it is principal, so we get an equation that cuts out the Weil divisor in a n.h. of this point. And a divisor cut out by a single .
More explicitly, a Cartier divisor is a choice of open cover U i of X, and meromorphic functions f i ∈ 𝒦 * (U i), such that f i / f j ∈ 𝒪 * (U i ∩ U j), along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if f i is replaced by .Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchangean open source textbook and reference work on algebraic geometry
Two Cartier divisors Dand D0are called linearly equivalent, denoted D˘D0, if and only if the di erence is principal. De nition 6.15. Let Xbe a scheme satisfying (). Then every Cartier divisor determines a Weil divisor. Informally a Cartier divisor is simply a Weil divisor de ned locally by one equation. If every Weil divisor is Cartier then we .A relative effective Cartier divisor is an effective Cartier divisor D ˆX such that the projection D !X is flat. We will show that this notion is well behaved under base-change by any S0!S. Lemma 1. Suppose D ˆX is a relative effective Cartier divisor for f : X !S. For any S0!S, denote by f0: X0!S0the pullback. Then D0= S0 S D ˆX0is a .More generally one can intersect a Cartier divisor with any subvariety and get a Cartier divisor on the subvariety, again provided the subva-riety is not contained in the Cartier divisor. Unfortunately using this, it is all too easy to give examples of integral Weil divisors which are not Cartier: Example 2.11.Cartier divisor on X, then it restricts to a closed subscheme on Y, locally cut out by one equation. If you are fortunate and this equation doesn’t vanish on any associated point of Y, then you get an effective Cartier divisor on Y. You can check that the restriction of effective Cartier divisors corresponds to restriction of invertible .
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this siteThe most useful divisors in algebraic geometry are the Cartier divisors, because they are intimately related to invertible sheaves and their sections. In thi.71.6 Effective Cartier divisors. For some reason it seem convenient to define the notion of an effective Cartier divisor before anything else. Note that in Morphisms of Spaces, Section 67.13 we discussed the correspondence between closed subspaces and quasi-coherent sheaves of ideals. Moreover, in Properties of Spaces, Section 66.30, we discussed properties of quasi .
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every Weil divisor is linearly equivalent to a Weil divisor supported on the invariant divisors, every Cartier divisor is linearly equivalent to a T-Cartier divisor. Hence, the only Cartier divisors are the principal divisors and Xis factorial if and only if the Class group is trivial. Example 3.6. The quadric cone Q, given by xy z2 = 0 in A3 k .
We study Cartier divisors on normal varieties with the action of a reductive groupG.We give criteria for a divisor to be Cartier, globally generated and ample, and apply them to a study of the local structure and the intersection theory of aG-variety.In particular, we prove an integral formula for the degree of an ample divisor on a variety of complexity 1, and apply this formula to .Two Cartier divisors Dand D0are called linearly equivalent, denoted D˘D0, if and only if the di erence is principal. De nition 2.3. Let Xbe a scheme satisfying (). Then every Cartier divisor determines a Weil divisor. Informally a Cartier divisor is simply a Weil divisor de ned locally by one equation. If every Weil divisor is Cartier then we . An effective Cartier divisor is actually a more directly geometric object, namely, it is a locally principal pure codimension one subscheme, that is, a subscheme, each component of which is codimension one, and which, locally around each point, is the zero locus of a section of the structure sheaf. Now in order to cut out a pure codimension one .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 principal if it is in the image of the map K∗(X) →K∗/O∗(X), and two Cartier divisors are linearly equivalent if their difference is principal. Write CaCl(X) for the group of Cartier divisors modulo linear equivalence. While this definition is abstract, think of it as follows: a Cartier divisor is a collection of Now this kind of divisor (i.e. locally principal divisor), is called the Cartier divisor, named after Pierre Cartier. Comparision of Cartier/Weil divisor: Cartier divisor gives up the extremely simple group structure of $\mathrm{\mathop{Div}}(X) .A Cartier divisor is principal if it is the divisor of a rational function i.e. div(r) where r ∈ R(X)∗. Two Cartier divisors differing by a principal Cartier divisor give rise to the same invertible sheaf. Rob told you that the Cartier divisor form an abelian group Div(X). When you mod How to define Cartier divisor and Weil divisor on algebraic stack? Do they correspond to line bundles on stack like the case of schemes? In case of a Deligne-Mumford stack, can we have a simpler
(*) In general, one may find Cartier divisors which cannot be written as a difference of effective Cartier divisors, which is the meat of the claim here. But since we are on an affine scheme, it trivially has an ample invertible sheaf and as discussed in Sasha's comment here this is enough.an open source textbook and reference work on algebraic geometryand get a Cartier divisor on the subvariety, again provided the subva-riety is not contained in the Cartier divisor. Unfortunately using this, it is all too easy to give examples of integral Weil divisors which are not Cartier: Example 2.11. Let XˆP3 be the quadric cone, which is given locally as X 0 = (xy z2) ˆA3. Then the line L= (x= z= 0 . Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange
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relative cartier divisor worksheet
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cartier divisor|locally principal divisor
