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.

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3 answers
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How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...
Theo Johnson-Freyd's user avatar
19 votes
3 answers
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What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
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18 votes
1 answer
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Embedding abelian varieties into projective spaces of small dimension

Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into. Is $d(A)$ uniform over all abelian varieties of a ...
Kim's user avatar
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12 votes
3 answers
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Modern treatment of Dirac monopoles and related topics

I know that the topic is classical and even "folklore", but many treatments make use of local coordinates and such treatments are rather messy. Could somewhere maybe provide some reference(s)...
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7 votes
2 answers
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Will (general points + small number of arbitrary points) impose independent condtions on plane curves?

It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
Drew's user avatar
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7 votes
1 answer
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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^*...
Oren's user avatar
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6 votes
6 answers
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What does the ample cone look like?

For a variety $X/k$, consider the monoid $A$ of classes of ample line bundles in $NS(X)$. What does $A \otimes_\mathbf{Z} \mathbf{R} \subset NS(X)_\mathbf{R}$ look like?
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5 votes
1 answer
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Identity relating iterated determinant line bundles

Suppose that $R$ is a (commutative, unital) ring and that $A$ is a (commutative, unital) $R$-algebra that is projective of constant rank $n$ as an $R$-module. Then $A$ has a "determinant line ...
Owen Biesel's user avatar
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3 votes
4 answers
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Cone over the Veronese surface

Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = \mathcal{O}_X(...
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3 votes
1 answer
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What are meromorphic line bundles?

Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it ...
მამუკა ჯიბლაძე's user avatar
3 votes
1 answer
474 views

Weak Lefschetz theorem for Lef line bundles

I'm studying M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772. The premises are the following....
Armando j18eos's user avatar
2 votes
1 answer
492 views

divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...
div's user avatar
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2 votes
0 answers
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Symmetric group-cocycle descends to symmetric product

Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
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1 answer
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Line bundles, linear systems and normalization

One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on $\mathbb{P}^...
Jodel's user avatar
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2 votes
2 answers
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Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
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2 votes
0 answers
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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 ...
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2 votes
2 answers
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Rational maps and Kodaira dimension

Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$. Assume that $Y$ is of general type. May we conclude then that $X$ ...
Puzzled's user avatar
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1 vote
2 answers
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What is the geometric point of view of an algebraic line bundle compared to a analytic line bundle?

Hi folkz, I'm trying to learn more about line bundles, invertible sheaves and divisors on schemes. I understand the connection beweteen Cartier and Weil Divisors and the connection between Cartier ...
Johannes's user avatar
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0 answers
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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 ...
user42066's user avatar
1 vote
3 answers
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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?
Messi's user avatar
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2 answers
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Line bundles with complex connection

Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...
Blake's user avatar
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