Divisor and line bundle
WebJan 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 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 …
Divisor and line bundle
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In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the sp… WebAn isomorphism of vector bundles over Xis then a morphism of vector bundles over Xsuch that that there exists an inverse (left and right inverse at the same time). Definition 1.8. …
WebThe 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 ((+)). 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. 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) …
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) = …
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. smosh park locationWebJun 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. smosh petsWebJul 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)) smosh paigeWebthere 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 … rizal time of birthhttp://math.stanford.edu/~vakil/0708-216/216class2829.pdf smosh people namesWeb1. 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 ... rizal took his last breakfast on earthWebJul 9, 2024 · 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. ( The degree of a line bundle L on a proper curve C over k is the degree of the divisor (s) of any nonzero rational section s of L.)A line bundle may also be called an invertible sheaf.. … rizal tower for rent