All Questions
Tagged with line-bundles reference-request
11 questions
3
votes
0
answers
136
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Relation between $\mathrm{Pic}^\natural_{X/S}$ and two notions of rigidification
Let $X/S$ be a relative curve (perhaps with more adjectives). I have come across a few instances of rigidifying and rigidificators, which I would like to understand better.
In Liu-Lorenzini-Raynaud (...
- 1,142
5
votes
1
answer
127
views
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 ...
- 2,356
1
vote
1
answer
324
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Reference request: $f^*D$ semi-ample, then $D$ semi-ample
I am looking for a suitable reference to put in a bibliography for the following fact:
Let $f: X \rightarrow Y$ be a surjective morphism between normal projective varieties. Let $D$ be a $\mathbb{Q}$-...
- 625
3
votes
0
answers
164
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Determinant of the universal bundle
Let $M$ be the moduli space of semistable vector bundles of fixed determinant $L$ and rank $r$ over a smooth curve $X$. Assume that $gcd(r,deg(L))=1$. Let $\mathcal U$ be the universal bundle over $M\...
- 1,891
4
votes
1
answer
309
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Nef line bundles over complex analytic spaces
Let $L$ be a line bundle over a compact complex manifold $X$ with a Hermitian metric $\omega$: $L$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth ...
- 828
2
votes
0
answers
160
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Universal property of limits of invertible sheaves
Let $R$ be a discrete valuation ring, $m$ the maximal ideal and $f:X \to \mathrm{Spec}(R)$ be a flat, proper morphism of relative dimension $1$. Assume further that $X$ is regular. For any $n>0$, ...
- 2,032
2
votes
2
answers
799
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Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?
Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...
- 9,019
0
votes
2
answers
435
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Isomorphism of connections on a complex line bundle
Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...
- 1,329
1
vote
0
answers
263
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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 ...
- 73
8
votes
1
answer
394
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Pullback along the Torelli map is an isomorphism
I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard ...
- 16k
6
votes
6
answers
5k
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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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