cartier divisor

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cartier divisor*******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 .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 .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 .cartier divisor regular local ring is ufdLearn 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 .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 .

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. .

Learn the definition and properties of effective Cartier divisors on schemes, which are closed subschemes cut out by a single nonzerodivisor. See examples, lemmas and .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.regular local ring is ufdA 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 . 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 .A Cartier divisor on a scheme Xis any global section of K=O X. In other words, a Cartier divisor is speci ed by an open cover U i, a collection of rational functions f i, such that f i=f j is a nowhere zero regular function. A Cartier divisor is called principal if it is in the image of ( X;K). Two Cartier divisors Dand D0are called linearly .

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 .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 .

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 site2 Cartier divisors When the scheme X is not regular, there is a more restrictive notion of divisors that turns out to be more useful in many cases. Let K be the locally constant sheaf associated to the function field K(X). A Cartier divisor on X is a section of the sheaf K(X)/O× . Using the construction of principal divisors,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 .Another partial way to handle Weil divisors is to restrict to $\mathbb Q$-Cartier divisors, i.e., a divisor that itself may not be Cartier, but a multiple of which is Cartier. Then you just take that multiple, pull that back and then divide by the same number you multiplied with.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 .an open source textbook and reference work on algebraic geometryThis lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.In this lecture we define W. 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

However, this procedure requires a lot of identification, and I wonder if there is a better characterization of the pullback of Cartier divisors. I've looked in some books but all I can find is the pullback of Weil Divisors. $\begingroup$ Restricting ourselves to the factorial case, where there is no Weil/Cartier distinction: effective divisors are cut out by regular sections of line bundles, while general divisors are cut out by rational sections. Also, I should maybe mention that something clicked in my head when I noticed that the Spanish term for cash is . This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.In this lecture we define W. 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 However, this procedure requires a lot of identification, and I wonder if there is a better characterization of the pullback of Cartier divisors. I've looked in some books but all I can find is the pullback of Weil Divisors. $\begingroup$ Restricting ourselves to the factorial case, where there is no Weil/Cartier distinction: effective divisors are cut out by regular sections of line bundles, while general divisors are cut out by rational sections. Also, I should maybe mention that something clicked in my head when I noticed that the Spanish term for cash is .(3) Is every Weil divisor on a singular curve $\mathbb{Q}$-Cartier (i.e. a mutiple of the divisor is Cartier)? A 2-dimensional analog seems to have been discussed in the following MO question: Is every Weil divisor on an arithmetic surface Q-CartierTwo 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 .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 .

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 .

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 .

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 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 .

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.

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cartier divisor|regular local ring is ufd