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
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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
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52
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Connection on line bundle over general simplicial toric variety
In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...
- 1,155
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1
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Picard group of toric varieties
I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...
- 1,155
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137
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Holomorphic line bundles associated to multiple U(1) groups, defined over toric manifolds
The sections of the holomorphic line bundle $\mathcal{O}(n)$ are acted on by the covariant derivative
$$
d+nA,
$$
where $A$ is the connection on the $U(1)$ bundle to which $\mathcal{O}(n)$ is ...
- 1,155
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2
answers
2k
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Square root of the determinant line
Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1_\Sigma$, denoted $\Omega^{1/2}_\Sigma$).
Let $\Gamma(\Omega^1_\Sigma)$ be the space of holomorphic ...
- 43.2k
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72
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Minimal non-klt center of asymptotic linear system
Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear ...
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2
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360
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The kernel of a nef line bundle
Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $...
- 13.8k
2
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1
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181
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Anti-canonical divisorial contractions of weak Fano $3$-folds
Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK_X|$ ...
user114198
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Holomorphic line bundles on smooth points of a quotient
I am an amateur algebraic geometer, so maybe this question is trivial and if this is the case, then I apologize. This is a question that came up while working on something completely different.
...
- 431
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In search for examples concerning pushforward of nef divisors and lc-trivial fibrations
My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...
- 625
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2
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924
views
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$ ...
- 8,998
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2
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968
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Rationality of conic bundles
Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.
Can ...
- 8,998
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1
answer
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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
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341
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Lefschetz type theorems for big and nef divisors
Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that the restriction map $Pic(X)\rightarrow Pic(D)$ is an isomorphism over $\mathbb{Q}$.
Under which ...
user82886
1
vote
1
answer
240
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Divisor class group of quartic surfaces
Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section.
What can we say about the singularities of $X$?
user97096
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525
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Lifting line bundles
Let $X$ be a smooth proper geometrically integral scheme over $\overline{\mathbb F_p}$. Assume $X$ is the specialization of a smooth proper scheme over $\mathbb Z_p^{nr}$. Let $L$ be an ample line ...
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172
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How close is $h^0(mD)$ to be a polynomial?
Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies.
At ...
- 625
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70
views
Moving curves and small transformations
Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...
- 141
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312
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Cone of moving curves
Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.
How can we detect when $C$ is an extremal ray ...
- 141
3
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2
answers
696
views
Fibers of pushforward of a bundle when the fiber dimension is not constant
I could not decide if I should post this question in MO or Mathstackexchange, so feel free to downvote it if you think it does not belong here. I will delete my post and post it in MathSE in that case....
- 325
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1
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273
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Big divisors in family
Given a family of divisors $D_t$ on varieties $X_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D_0$ ...
- 625
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200
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Top intersections on the Hilbert scheme of points on a surface
The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...
- 1,509
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1
answer
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Pull-back of the canonical divisor via a rational map
Let $f:X\dashrightarrow Y$ be a birational map between projective varieties not contracting any divisor. Assume that $X$ is smooth, and that $Y$ has at most ordinary singularities at finitely many ...
user97096
1
vote
1
answer
242
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Number of conditions imposed by fat points to a linear system
Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...
user97096
0
votes
1
answer
192
views
Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem?
I try to understand the following version of the Kodaira embedding theorem:
Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample.
I have a problem with the '...
- 9
1
vote
1
answer
210
views
Hermitic connections on complex line bundles with imaginary curvature form
It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
- 5,407
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241
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Moment map of equivariant line bundles
I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...
- 111
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1
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What is known about the cohomology of the relative tangent bundle on a conic bundle?
Let us assume $X$ is a smooth, projective and unirational variety of dimension $n$ over $\mathbb{C}$.
Given a conic bundle $\pi: Y\rightarrow X$ such that $\omega_{\pi}^{-1}$ is relatively very ample ...
- 1,025
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1
answer
511
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Rigid effective divisors
Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let $f:X\...
user77919
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179
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How the existence of holomorphic sections depends on the choice of complex structure
In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...
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1
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308
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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 ...
- 502
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answer
575
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Pushforward of line bundle under "toric isogeny"
Let $(X,T)$ be a smooth complex toric variety of dimension $d$ with torus $T$ and toric boundary $D=X\setminus T$. Let $\phi : X\to X$ be a finite endomorphism of $X$ such that the restriction
$$\phi|...
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Line bundles with vanishing cohomology on Calabi-Yau manifold
Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...
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Picard groups, ample cones, and proper birational maps
Let $f:Y\to X$ be a proper birational map of normal varieties over an algebraically closed field which is an isomorphism over the regular locus.
Q1: Is it the case that the pullback $f^*\...
- 2,537
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0
answers
160
views
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
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0
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152
views
Are two line bundles with the same ramification type necessarily isomorphic?
I have no motivation for the following problem, I am just curious if it is true or not. Here it is:
If $l_1$ and $l_2$ are two complete $g^r_d$'s on a smooth curve $C$ such that the vanishing ...
- 325
2
votes
2
answers
799
views
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
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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 ...
- 1,509
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1
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421
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Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points
Let $S^{[n]}$ be the Hilbert scheme of $n$ 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
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Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial? [closed]
Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample?
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Movable Divisors
Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?
user56259
3
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4
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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(...
user56259
2
votes
1
answer
422
views
How to see that this pairing of line bundles is multiplicative?
Given a projective flat morphism $p: X \rightarrow Y$ of integral noetherian schemes of relative dimension one.
For a coherent sheaf $F$ on $Y$ we can define a line bundle $det(F)$ on $Y$ and for a ...
- 251
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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
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0
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251
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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 ...
- 1,509
2
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2
answers
526
views
Holomorphic Line Bundles over a Homogeneous Space
Let $M=G/H$ be (compact) homogeneous complex manifold, and let $L$ be a line bundle over $M$. Can one always equip $L$ with a holomorphic structure? Can there be more then one such holomorphic ...
1
vote
1
answer
188
views
Perfectness of the Jacobian of a curve
Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer $r>1$....
- 503
20
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1
answer
2k
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Maps to projective space == line bundles; what do maps to weighted projective space correspond to?
A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective space?...
- 40.3k
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1
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248
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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
3
votes
2
answers
5k
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Big and Nef divisors
In Example 2.2.19 of
Lazarsfeld, Positivity in Algebraic Geometry I,
I found the following statement:
Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and ...
user47036