Divisor and line bundle

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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). Web1. Degree of a line bundle / invertible sheaf 1.1. Last time. Last time, I de ned the Picard group of a variety X, denoted Pic(X), as the group of invertible sheaves on X. In the case when X was a nonsingular curve, I de ned the Weil divisor class group of X, denoted Cl(X)= Div(X)=Lin(X), and sketched why Pic(X) ˘=Cl(X). Let me remind you of ...

Line bundles: from transition functions to divisors

WebJun 3, 2016 · Sorted by: 5. This holds for any proper scheme over k, since the set of all such effective divisors is in bijection with ( H 0 ( X, L) − { 0 }) / k ∗. See chapter II.7 in Hartshorne's Algebraic geometry, in particular the part about linear series. Share. thick leather wow

Line bundle - Wikipedia

http://math.stanford.edu/~vakil/0708-216/216class2829.pdf 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 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 ... thick leather office chair

The canonical divisor - University of California, San Diego

algebraic geometry - Weil divisors on normal Noetherian schemes ...

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 … 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) = … thick leather strap watches miami flWebThe Birkhoff-Grothendieck theorem states that on the projective line, any vector bundle splits in a unique way as a direct sum of the line bundles. Important line bundles. The tautological bundle, which appears for instance as the exceptional divisor of the blowing up of a smooth point is the sheaf (). The canonical bundle (), is ((+)). sai international school gadhinglaj

"WebIn 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. " - Divisor and line bundle

Divisor and line bundle

Line bundle associated to a divisor - Purdue University

WebRiemann–Roch for line bundles. Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X. Let (,) denote the space of holomorphic sections of L. WebLinear 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 …

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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 … 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.

WebLinear 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 bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic. Examples Linear equivalence 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

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 … 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 …

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

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. sai international school careerWeb1. 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) … sai international school codeWebWeil divisors and rational sections of line bundles need not hold. So, to get a nicely behaved theory of divisors on these more general schemes, we apply the \French trick … sai international school bhubaneswar websiteWebThe 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) … sai international school bhubaneswar addressWebthere 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 … sai international school unwindWeb1. 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. sai international school - bhubaneswar emailWebJan 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 thick leathery skin on legs