Questions tagged [ample-bundles]
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53
questions
4
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One-point compactification of ample line bundle
Given a smooth complex projective variety with an ample line bundle $L$, it seems to be folklore that one can get a one-point compactification of the total space $\mathbb{V}(L)$ of $L$ such that ...
- 151
1
vote
0
answers
133
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GAGA, positive line bundles, Kodaira embedding, and homogeneous coordinate rings
Let $M$ be a compact K"ahler manifold and let $L$ be a positive line bundle over $M$. We know from the Kodaira embedding theorem that from $L^{\otimes k}$ for some $k$ we can construct an ...
- 101
1
vote
1
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300
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Making a vector bundle ample by twisting with ample line bundle
Let $X$ be a projective algebraic variety over some field (I am happy to add some more assumptions if necessary). A vector bundle $E$ is ample if the relative twisting sheaf $\mathcal{O}_{\mathbf{P}(E)...
user279422
7
votes
0
answers
220
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Ample divisors on $T$-varieties
Question: how does one use a torus action to help decide whether a divisor or line bundle is ample?
In more detail: I have a (normal, but usually rather singular) $T$-variety $X$, where $T$ is an ...
- 5,672
4
votes
0
answers
231
views
How much does the formal completion know about the ambient variety?
How much information does the formal completion along an ample divisor hold about the ambient variety? Given two smooth projective varieties such that they have isomorphic ample divisors, where the ...
- 5,607
3
votes
0
answers
148
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Functorial lift of certain vector bundles to the ambient projective space
Given an very ample line bundle $L$ on a projective variety we embed it into a projective space such that pullback of $\mathcal{O}(1)$ is $L$. Then we can identify the two. Consider the full sub-...
- 5,607
2
votes
1
answer
334
views
Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$
Let $p:C\to\mathbb{P}^1$ be a degree $k$ morphism from a smooth projective curve $C$ to the projective line and $L$ a very ample line bundle on $C$. We know that $p_*\mathcal{O}_C(L)$ is a rank $k$ ...
- 355
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0
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325
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Embedding Calabi-Yau manifolds in projective space
When studying homological mirror symmetry, a lot of work is done not in the setting of complex manifolds, but of smooth (quasi-)projective varieties, see e.g. a paper from Orlov. However, the actual ...
- 821
2
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0
answers
98
views
Nef and effective cone of minimal conic bundle
Let $\pi: S\to C$ be a minimal conic bundle over a field $k$ of characteristic zero. That is, $S$ is a geometrically irreducible smooth surface with Picard rank $2$ and $C$ a geometrically ...
- 2,721
2
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139
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Structure of the big cone and Seshadri constant on Fano manifolds
I would like to know something about the following two questions.
Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define
\begin{gather}
\sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\,...
- 71
1
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0
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144
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Ample line bundle gives alternative description of a variety
Let $X$ be a (smooth) projective variety (over $\mathbb{C}$), and $\mathcal{L}$ an ample line bundle on $X$. I have heard that then
$$ X \cong \mathrm{Proj} \left( \bigoplus_{k \ge 0} H^0(X,\mathcal{...
- 513
2
votes
1
answer
385
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Classifying ample line bundles over the flag manifold $G/B$
For a complex Lie group $G$, with $B$ a choice of Borel subgroup. The line bundles over the flag manifold $G/B$ are indexed by elements of the weight lattice of $\frak{g}$. Which of these line bundles ...
2
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0
answers
241
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Twisting holomorphic vector bundles and Euler characteristics
Given a holomorphic vector bundle $\mathcal{V}$ over a compact complex manifold $M$, it seems that even if $\mathcal{V}$ is non-trivial, then it can still have trivial Euler characteristic, that it,
$$...
- 633
2
votes
1
answer
809
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Bertini type theorem for very ample line bundle
Let $X$ be a normal, projective variety (can take $X$ to be a hypersurface in a projective space) of dimension at least $3$. Let $L$ be a very ample line bundle on $X$, hence base-point free. What can ...
- 2,002
5
votes
1
answer
772
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Ample vector bundles and embeddings
If $X$ is a complete variety over a field, a line bundle $L$ is said to be very ample if there is a closed immersion from $X$ into a projective space, such that the pullback of $\mathcal{O}(1)$ is ...
- 401
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131
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Global sections of ample line bundles over (rational) elliptic fibration
Let $S$ be a smooth, complex elliptic fibration over $\mathbb{P}^1$ and $L$ be an ample invertible sheaf on $S$. I am looking for criterion under which $L$ has a non-trivial global section. Any idea/...
- 2,096
3
votes
1
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112
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The ample cone of a surface with an algebraic $\mathbb C^*$ action
Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder ...
- 13.8k
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1
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274
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Does a projective variety have only finitely many associated Hilbert polynomials?
Let $X$ be a projective variety over $\mathbb{C}$. If $L$ is an ample line bundle, then $h_L$ denotes the Hilbert polynomial.
Is it true that, if $L$ and $L'$ are ample line bundles which are ...
- 43
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Does ampleness descend along finite maps?
First, let me emphasize that for $X$ a not-necessarily proper variety, we say that a line bundle $L$ on $X$ is ample, if for some positive integer $n$, $L^{\otimes n}$ arises as $j^*O(1)$ for some (...
- 2,716
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0
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249
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Very ample divisors on blow ups of the projective plane
Let $X$ be $\mathbb{P}^2$ blown up at $k$ points in general position. The Picard group of $X$ is just $\mathbb{Z}^{k+1}$ and one knows the intersection product explicitly. If $D$ is an ample divisor, ...
- 2,721
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0
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81
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$H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?
Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.
Is there a constant $C=...
- 625
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1
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596
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Fiber product of projective varieties and ample line bundles
Let $X, Y$ be smooth, projective varieties, $L_X$ and $L_Y$ are very ample line bundles on $X$ and $Y$, respectively. If I understand correctly, $\mbox{pr}_1^*L_X \otimes \mbox{pr}_2^*L_Y$ is a very ...
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112
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Do integral curves on simple abelian surfaces define big line bundles?
Let $A$ be a simple abelian surface over $\mathbb{C}$.
Let $C\subset A$ be an irreducible and reduced one-dimensional closed subscheme. Since $A$ is simple, the normalization of $C$ is of genus at ...
- 57
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559
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Hartshorne's Conjectures about Algebraic Bundles?
In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.
I understand that many of these ...
user100272
4
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0
answers
246
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When is the strict transform of very ample divisor ample?
Let $X$ be a projective variety and $Y \subset X$ is a very ample divisor on $X$. Let $Z \subset X$ be a regular subvariety (but $X$ need not be regular along $Z$) of codimension $2$ and $\pi:\tilde{X}...
- 2,096
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vote
0
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137
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Does being big for a line bundle satisfy fpqc descent
Let $k$ be a field of characteristic zero, and let $L/k$ be a field extension. [Assume $k$ and $L$ are algebraically closed if necessary.]
Let $X$ be a variety over $k$ and let $\mathcal{L}$ be a ...
- 19
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280
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What is the support of a coherent sheaf on $X\times Y$ if it is invariant by tensoring a very ample line bundle on $X$?
Let $X$ be a smooth projective variety over a field $k$ with char$k=0$ and $\mathcal{L}$ be a very ample line bundle on $X$. Let $\mathcal{F}$ be a coherent sheaf on $X$. It is well-know that if $\...
- 8,417
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Extension of ample vector bundles is ample
As I read Huybrechts-Lehn's book on Moduli of Sheaves, it is making a claim that extensions of several (at least 2) ample vector bundles (on curves) is again ample. Somehow, I am unable to see this ...
- 159
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1
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293
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Pencils in very ample linear systems without curve in its base locus
If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...
- 482
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3
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1k
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Varieties with an ample vector bundle mapping to their tangent bundle
A well-known result of Andreatta and Wisniewski says: Let $X$ be a projective complex manifold whose tangent bundle $T_X$ contains an ample sub-bundle $\mathscr{E}$. Then $X$ is isomorphic to ...
- 21.7k
2
votes
4
answers
2k
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Global section of very ample line bundles and its value on stalks
Let $X$ be a projective scheme and $\mathcal{L}$ be a very ample line bundle on $X$ with respect to some projective embedding $X \hookrightarrow \mathbb{P}^n_{\mathbb{C}}$ (for some $n$).
Given any ...
- 1,909
11
votes
2
answers
854
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On a proposition in Hartshorne's paper "Ample vector bundles on curves"
In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field $...
- 2,986
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0
answers
234
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Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR
Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...
- 493
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votes
1
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230
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Big divisors and small transformations
Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that:
$-K_Y$ is ...
user49214
1
vote
1
answer
229
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Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points
Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let
$$D = aH-b_1E_1-...-b_kE_k$$
be a divisor on $X$. Are there conditions on $a,b_1,...,b_k$ ensuring that $...
user49214
3
votes
0
answers
229
views
Ampleness of Hodge bundles over complex curves
Let $C$ be a smooth, proper and connected curve over
the complex numbers $\bf C$. Let ${\cal G}\to C$ be a smooth group scheme over $C$ and let $\epsilon_{\cal G}:C\to{\cal G}$ be its
zero-section. ...
- 2,986
4
votes
1
answer
562
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Does normalization of projective varieties preserve very ampleness
Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any ...
- 823
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Weak Fano and Log fano varieties
A projective smooth variety $X$ is weak Fano if $-K_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K_X+D)$ is ample and $(X,D)$ is Kawamata log terminal.
Is ...
user49214
1
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2
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490
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An ample line bundle on a K3 surface
Let $X$ be a K3 surface obtained as a double covering of $\mathbb{P}^1 \times \mathbb{P}^1$ branching along a $(4,4)$-divisor. I think the natural line bundle $\pi^*\mathcal{O}_{\mathbb{P}^1\times \...
- 23
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1
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2k
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Kleiman's and Nakai-Moishezon's ampleness criteria
I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion.
Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at ...
user47036
1
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1
answer
291
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A question on very ample line bundle on smooth projective surfaces
I had been reading a couple of texts by J.P. Demailly, one of them titled "Effective bounds for very ample line bundles". In the introduction the author mentions a result due to I. Reider (stated in "...
- 823
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votes
1
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509
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Canonical bundle of the moduli space of curves
By the pointed canonical bundle formula the canonical bundle of $\overline{M}_{g,n}$ is given by
$$K_{\overline{M}_{g,n}} = 13\lambda+\psi-2\delta-\sum_{I}\delta_{1,I}$$
where $\lambda$ is the Hodge ...
- 6,770
3
votes
0
answers
330
views
ampleness in families
Let $X\to S$ be a smooth projective morphism with geometrically connected fibres over an integral noetherian regular scheme $S$ with generic point $\eta$.
Let $L$ be a line bundle on $X$, and suppose ...
- 31
4
votes
1
answer
274
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Extension of linear system
Let $X$ be a normal irreducible closed subvariety of $\mathbb{P}^n$ and let $\Lambda$ be a linear system on $X$, without fixed component. Then, there exists a linear system $\Lambda'$ on $\mathbb{P}^n$...
- 7,415
1
vote
1
answer
493
views
Does every ample divisor "span" a hyperplane?
Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...
- 257
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0
answers
521
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(Relative) ampleness on algebraic spaces
This is a follow-up (of sorts) to this question.
Let $f : X \to T$ be a proper morphism of schemes. Then the notion of a relative ample (or $f$-ample) line bundle can be defined in several ...
- 884
3
votes
0
answers
176
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Ampleness on the P^1 bundle over Siegel threefold
I am looking at the Shimura variety for $\mathrm{GSp}_4(\mathbb Q)$, with hyperspecial level structure at $p$. Let $X$ denote the special fiber over $\mathbb F_p$. For simplicity, let us pretend ...
- 61
1
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1
answer
1k
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When is an ample line bundle on an abelian variety base point free?
So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed ...
- 15.6k
4
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2
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1k
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When is the determinant of the push-forward of an ample line bundle ample
Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$.
When is $\det f_\ast L$ also ample?
A "nice" morphism could be anything from "finite type separated" to ...
0
votes
0
answers
702
views
About first Chern class and Poincaré duality in case of an ample divisor
Led $D$ be a very ample divisor in $X$ projective variety.
I can't understand why the first Chern class $c_1(\mathscr{O}_X(D))$ equals the Poincaré dual of $D$, $\mathscr{P}(D)$
- 107