Questions tagged [line-bundles]
A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.
231 questions
6
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2
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What kind of line bundles have Chern class of Hodge type (2,0) or (0,2)?
If $L$ is a complex line bundle on a topological space $X$, let $c_1(L)$ denote the image of its Chern class in $H^2(X;C)$. A complex manifold structure on $X$ [ok which is also compact and say ...
- 4,949
8
votes
1
answer
952
views
volume of big line bundles under finite morphisms
Let $X$, $Y$ be complex projective varieties of dimension $n$, let $f:X \rightarrow Y$ be a surjective finite morphism of degree $d$ and let $B$ be a big line bundle on $Y$.
Is that true that vol($f^*...
- 205
0
votes
0
answers
183
views
When can one find holomorphic sections vanishing at a point to a certain order?
Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...
- 3,245
2
votes
0
answers
251
views
Computing Euler Charactistics of Line bundles on Hilbert Schemes of points on Surfaces
Let $S^{[2]}$ be the Hilbert scheme of two points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the ...
- 1,509
0
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1
answer
140
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Two questions about line bundles over Kuranishi families
i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too ...
- 63
6
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1
answer
2k
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Line bundles: from transition functions to divisors
Recently I was thinking about how local systems are the same thing as vector bundles with flat connection, and how representations of the fundamental group gave rise to vector bundles. This got me ...
- 2,168
1
vote
3
answers
999
views
On the Clifford index of a curve
Let X be an algebraic curve and c be the Clifford index of X.
When c is small (e.g c=1), what is the classification of the line bundle who computes c?
- 11
3
votes
3
answers
687
views
Nature of Invertible Sheaves in which there are no global sections.
EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
- 2,132
3
votes
2
answers
226
views
Uniformity of injectivity for maps associated to linear systems
Let $X$ be a compact complex manifold and $L\to X$ a holomorphic line bundle (without any a priori assumption on its positivity).
Suppose that for each $x,y\in X$, with $x\ne y$, there exists a $k_0\...
- 7,902
1
vote
0
answers
389
views
canonical model of a reducible curve
Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...
- 3,779
1
vote
0
answers
263
views
Hopf lemma for line bundles on curves in algebraic geometry
In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...
- 73
0
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0
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200
views
canonical bundle of the relative spectrum
maybe it is a very trivial quetion but:
suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum $Spec(Sym(\mathcal{A}))\...
- 1
5
votes
1
answer
630
views
Line bundles in abelian $\otimes$-categories
By an abelian $\otimes$-category I mean a symmetric monoidal category $(\mathcal{A},\otimes,\mathcal{O})$, such that $\mathcal{A}$ also is an abelian category and for every $M \in \mathcal{A}$ the ...
- 63.1k
5
votes
4
answers
667
views
Sections of a divisor on elliptic curve
I'm interested in producing explicit bases for the sections of a line bundle on an embedded genus 1 curve. Let me restrict to the first case that I don't know how to do, so that I can be as concrete ...
- 2,955
1
vote
0
answers
441
views
Theta functions and Fourier transforms
Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...
- 1,025
5
votes
1
answer
527
views
graded ring associated to a line bundle in a tensor category
Let $\mathcal{A}$ be an abelian tensor category with unit $\mathcal{O}$. An object $\mathcal{L}$ is called invertible or a line bundle if there is some $\mathcal{L}^{-1}$ such that $\mathcal{L} \...
- 63.1k
1
vote
2
answers
470
views
Connections with compatible Hermitian products on complex line bundles
Let $X$ be a manifold, $L$ be a complex line bundle over $X$, and $L^{*}$ be the associated principal bundle. Suppose $\alpha$ is a connection form on $L^{*}$, with associated connection $D$ on $L$. ...
- 1,025
4
votes
1
answer
1k
views
Tensor product of a line bundle with a large multiple of another positive line bundle also positive?
Let $X$ be a complex manifold and $\mathcal{L}$ be a positive line bundle on $X$. If $E$ is any other line bundle on $X$, then is it true that for all sufficiently large $m$, $\mathcal{L}^m \otimes E$ ...
- 5,359
2
votes
1
answer
424
views
Different ways to construct maps and the tensor products of line bundles
Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms from $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective ...
- 16k
1
vote
1
answer
129
views
Unicity of a vector field on $S^1$-bundle
Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...
user21574
1
vote
1
answer
373
views
etale covers of line bundles on an abelian variety
subj: etale covers of line bundles on an abelian variety
Is there an explicit decryption of finite
etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles
$L^o = L \...
- 468
3
votes
1
answer
700
views
Pulling back a line bundle on the Jacobian to a spin bundle on the curve
I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta ...
4
votes
1
answer
988
views
Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces
I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a globally ...
- 30.6k
1
vote
0
answers
259
views
How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
Who knows references about this?
In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
- 997
4
votes
1
answer
1k
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Line Bundles on Torus Quotient
Suppose you have a scheme $X$ that is acted on by a torus $T$. Then the action induces a grading on the functions on $X$ by the character lattice of $T$. So for a fixed character $\lambda$, we can ...
- 5,548
2
votes
1
answer
263
views
Followup; Strict Transform of a Line Bundle
This is a follow up to my previous question, and I have lowered my demands to a situation as follows:
Let $X$ be an algebraic variety, $\mathcal{I}$ a coherent sheaf of ideals and $\mathcal{L}$ a ...
- 4,902
10
votes
0
answers
526
views
Deformations of some simple quotient stacks.
I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.
I will ...
- 1,180
6
votes
0
answers
367
views
Do simplicial toric varieties have "lots" of base point free linear systems?
Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system?
If $X$ is not ...
- 24.1k
1
vote
0
answers
138
views
pairing theta functions for different complex structures
I apologize for my previous attempt to ask this, which was very badly written.
Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex ...
- 1,025
0
votes
0
answers
251
views
Does the normalization of a projective morphism determine the line bundle?
Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms
$$f:X \to \mathbb{P}^n$$
and
$$g:X \to \mathbb{P}^m,$$
such that the image of $f$ is the ...
- 1
3
votes
0
answers
226
views
How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?
Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...
- 3,779