2024 Divisor and line bundle

Divisor and line bundle

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WebAG 5 2. Meromorphic functions, divisors and line bundles Let Xbe a smooth algebraic variety, i.e., Xis holomorphically em-bedded in some Pn. let Fand Gbe two homogeneous polynomials over Pn of the degree d. Consider the quotient Web3 This implies that ord p(f!) = nk+ n 1 = (ord f( )!+ 1)e p(f) 1 which proves our assertion. Proposition 1.1. Let Xbe a compact Riemann surface of genus gand K X be a canonical divisor. Then degK X = 2g 2: Proof. Let fbe a nonconstant meromorphic function on X:Then f: X!P1 is a noncon- stant holomorphic mapping and thus a rami ed covering of …

Nef line bundle - HandWiki

Web1. Line bundle associated to a divisor Given a divisor D= P n pp, recall that we can associate a sheaf O X(D). By construction, when Uis a coordinate disc, O(D)(U) = O X(U) … WebIn brief: one has two different ways of regarding line bundles on a smooth complex algebraic variety, as a set of transition functions and as an equivalence class of Weil … january border clipart

algebraic geometry - Riemann-Roch on vector bundles with divisors …

Webabove, and deform the divisor to a linearly equivalent divisor, which does not contain the curve. A more sophisticatedapproach is as follows. If the image of the curve lies in the divisor, then instead of pulling the divisor back, pullback the associated line bundle and take the degree of that D · f C = degf∗O X(D). Deﬁnition 2.13. Webthere is a divisor D02jmDj, not containing x. But then kD02jkmDj is a divisor not containing x. Pick m 0 such that H0(X;O X(mD)) O X! O X(mD); is surjective for all m m 0. Since … WebMar 6, 2024 · Every line bundle L on an integral Noetherian scheme X is the class of some Cartier divisor. As a result, the exact sequence above identifies the Picard group of line bundles on an integral Noetherian scheme X with the group of Cartier divisors modulo linear equivalence. lowest temp for pothos

Line bundle associated to a divisor - Purdue University

Divisor -- line bundle correspondence in algebraic geometry

WebTheorem 13. A divisor and a meromorphic section of a holomorphic line bundle are essentially the same thing. More precisely (i) Every holomorphic line L!Xadmits a … WebMar 21, 2024 · The map div ( s): P i c ( X) → C l ( X) is injective. This means for any normal noetherian scheme, any Cartier divisor gives a Weil divisor. We see that the reverse is not always true: the canonical example is the divisor D given by the line V ( x, z) inside the cone V ( x y − z 2) ⊂ A 3 (Vakil exercise 14.2.H). january border ideasWeb1. Invertible sheaves and Weil divisors 1 1. INVERTIBLE SHEAVES AND WEIL DIVISORS In the previous section, we saw a link between line bundles and codimension 1 infor-mation. We now continue this theme. The notion of Weil divisors will give a great way of understanding and classifying line bundles, at least on Noetherian normal schemes. lowest temp for paint

"Webdivisor, Lfor line bundle. Xprojective. Recall that there are many ways of de ning ampleness for line bundle L: (1) some large power is very ample, (2) cohomological … " - Divisor and line bundle

Divisor and line bundle

Line bundles: from transition functions to divisors

WebEffective divisors correspond to line bundles with nontrivial holomorphic sections, then given a line bundle you can just choose any holomorphic section. Its divisor of zeroes … Webisomorphism L!L0of holomorphic line bundles which carries sto s0. (v) Two divisors are linearly equivalent if and only if the corresponding holo-morphic line bundles are isomorphic. (vi) Let Dbe the divisor of a meromorphic section sof a holomorphic line bundle L!X. Then the map L(D) !O(X;L) : f7!fs

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WebThe question of whether every line bundle comes from a divisor is more delicate. On the positive side, there are two suﬃcient conditions: Example 1.1.5. (Line bundles from divisors). There are a couple of natural hypotheses to guarantee that every line bundle arises from a divisor. (i). If Xis reduced and irreducible, or merely reduced, then ... Webabove, and deform the divisor to a linearly equivalent divisor, which does not contain the curve. A more sophisticatedapproach is as follows. If the image of the curve lies in the …

WebA complex line bundle is a 2 dimensional vector bundle with a complex structure on each fiber, i.e. each change of coordinates \( g_{ij}: ... 1.2 Divisors, line bundles and sheaves. A holomorphic line bundle is the same as a locally free \( \mathcal{O}_X \)-module of rank 1.

WebRecall that by DivXwe denote the group of divisors, and there is no ambiguity in this notion if Xis a smooth projective variety. Recall also that if Dis a divisor, then we can associate a line bundle to it, and this line bundle is denoted by O X(D). Theorem 1.2.1. Let Xbe a smooth projective surface. Then there is a unique pairing

Weba divisor D= (fU ;f g), de ne a line bundle L= O(D) to be trivialized on each U with transition functions f =f . Two Cartier divisors Dand D0are linearly equivalent if and only if O(D) = … lowest temp for marigoldsWebJan 8, 2015 · 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 site january books for toddlersWebJul 3, 2024 · 1. Let X be a Riemannian surfaces with a divisor D and let E be a holomorphic complex vector bundle of rank r on X. 1) The Riemann-Roch theorem is used to give an estimate of the dimension of the vector space of the holomorphic sections of E, i.e. dim ( H 0 ( X, E)) − dim ( H 1 ( X, E)) = deg ( E) − r k ( E) ( 1 − g ( X)) lowest temp for potted cactusWebThe Divisor-Line Bundle Correspondence So we have a injective homomorphism f(L;s)g=(X;O X)! Cl(X) We can construct an inverse: Let D be a Weil divisor and let L(D) … january bookshelf pdfWebLinear systems can also be introduced by means of the line bundle or invertible sheaf language. In those terms, divisors (Cartier divisors, to be precise) correspond to line … lowest temp for pot plantsWebDe nition 2. We have noted before that isomorphism classes of line bundles over a scheme for a group, with the ring of regular functions as the identity, tensor product as the operation, and dualizing as the inverse. We call this group the Picard group of a scheme X, or Pic(X). Lemma 2 (Line Bundles are Cartier Divisors). There is a natural ... january border clip art freeWebIn view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. Definition. More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X. lowest temp for outdoor painting