# 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.

55
questions with no upvoted or accepted answers

**10**

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498 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 ...

**6**

votes

**0**answers

748 views

### 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 ...

**6**

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338 views

### Do line bundles which divide the canonical bundle lift

Let $S $ be the spectrum of a complete dvr with algebraically closed residue field. Let $\eta$ be its generic point and let $s$ be its closed point with $k(s)$ of positive characteristic. Let $X\to S$ ...

**6**

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360 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 ...

**5**

votes

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104 views

### Does there exist a notion of Chern classes in intersection cohomology?

First of all: I apologize for my mistakes, I'm a freshman in intersection cohomology.
Let $X$ be a (compact) complex analytic space, let $L$ be a line bundle over $X$.
Can one define a notion of ...

**5**

votes

**0**answers

173 views

### 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 ...

**4**

votes

**0**answers

104 views

### Arnold's theorem on small denominators and holomorphic tubular neighborhoods

By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...

**4**

votes

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154 views

### Does there exist a preferred trivialization of a trivial line bundle?

Let $L\to M$ be a topologically trivial complex Hermitian line bundle (over a manifold of dimension three, if this is of any importance). I assume that $L$ admits a trivialization, however, I do not ...

**4**

votes

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161 views

### 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 ...

**3**

votes

**0**answers

36 views

### Isometries of fiber bundles

Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...

**3**

votes

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107 views

### A question on Okounkov bodies

Let $X$ be an irreducible $n$-dimensional projective variety, and
$$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$
a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...

**3**

votes

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129 views

### Is Wronskian a line bundle for Riemann surfaces?

Suppose $f_1,\dots,f_g$ are holomorphic functons on a domain $U\subset\mathbb{C}$. By the Wronskian determinant $f_1,\dots,f_g$ one means the determinant of the matrix of derivatives $f_k^{(m)},$ ...

**3**

votes

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141 views

### 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\...

**3**

votes

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251 views

### 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 ...

**3**

votes

**0**answers

167 views

### 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 ...

**3**

votes

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148 views

### 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 ...

**3**

votes

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553 views

### Ample Line Bundles on Algebraic Spaces

The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...

**3**

votes

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218 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 ...

**2**

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**0**answers

137 views

### What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?

Let $X$ be a projective homogeneous space over $\mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $f\colon\mathbb{P}^1\to X$ ...

**2**

votes

**0**answers

74 views

### The dual of the space of continuous sections in a vector bundle

If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...

**2**

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**0**answers

64 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 ...

**2**

votes

**0**answers

125 views

### 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.
...

**2**

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197 views

### 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 ...

**2**

votes

**0**answers

155 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**

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153 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 ...

**2**

votes

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136 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**

vote

**0**answers

103 views

### Proof of uniqueness in the universal property of Poincaré line bundles

My question concerns the proof of a part of Lemma IV.2.2 (pag. 168) of the book Geometry of Algebraic Curves. vol. I by Arbarello, Cornalba, Griffiths and Harris. In order to state my problem, let me ...

**1**

vote

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69 views

### Discriminant divisor $\mathcal{D}_{r} \subseteq H^{0}(X,K_{X}^{\otimes r})$ is irreducible

Let $X\colon$ smooth projective curve over $\mathbb{C}$, $K_{X}\colon$ canonical line bundle over $X$, and $W_{r}$ denotes $H^{0}(X,K_{X}^{\otimes r})$.
I'm trying to prove the following proposition,
...

**1**

vote

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78 views

### Vector bundle defined by using divisors of very ample line bundle

Let $X$ be a smooth projective curve. Suppose that $L_1$ and $L_2$ are line bundles on $X$, and $L_1$ is very ample.
$\operatorname{Div}(s)$ denotes a divisor defined by a global section $s\in H^0(X,L)...

**1**

vote

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120 views

### Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$

Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...

**1**

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103 views

### Problem regarding existence of a divisor representing line bundle

We consider a normal irreducible variety $X$ and a line bundle $L$. The question is when $L$ is induced by a Cartier divisor $D$. We know that if $s$ is a rational section of $O_X(D)$, where $D$ is a ...

**1**

vote

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89 views

### The group of global sections of the automorphism bundle of the tangent bundle on a Grassmannian

Let $X={\rm Gr}(k,n)$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb C}^n$.
We regard $X$ as an algebraic variety over $\Bbb C$.
Let ${T_X} \to X$ denote the tangent bundle on $X$. For ...

**1**

vote

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185 views

### Splitting principle in algebraic geometry and ample line bundles

Splitting theorem in algebraic geometry claims that if we have a vector bundle $V$ on $X$ (we consider a smooth projective variety for this question), if we pull-back $V$ to $\mathbb{P}(V)$, we get a ...

**1**

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**0**answers

129 views

### 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{...

**1**

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**0**answers

133 views

### Are torsion-free rank 1 modules over integral schemes line bundles?

How far away are torsion-free rank 1 sheaves from the line bundles? Is there any condition that makes sure they are same? (for dimensions higher than 1). It is known that for a regular scheme of ...

**1**

vote

**0**answers

156 views

### Torsion line bundle on hyperelliptic curves and Weierstrass points

Let $C$ be an hyperelliptic curve of genus $g$ and let $f:C\rightarrow
\mathbb{P}^1$ be the corresponding 2 to 1 covering
ramified in $2g+2$ points.
Let $L$ be a line bundle on $C$ such that either $...

**1**

vote

**0**answers

106 views

### Twisting a line bundle with the zero section

Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero ...

**1**

vote

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47 views

### 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**

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**0**answers

110 views

### 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**

vote

**0**answers

51 views

### 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 ...

**1**

vote

**0**answers

95 views

### 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.
...

**1**

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**0**answers

149 views

### 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 ...

**1**

vote

**0**answers

140 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 ...

**1**

vote

**0**answers

201 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 ...

**1**

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**0**answers

339 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) ...

**1**

vote

**0**answers

240 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 $R$ ...

**1**

vote

**0**answers

374 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 ...

**0**

votes

**0**answers

107 views

### Sections of vector bundles interpreted as sections of line bundles

Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...

**0**

votes

**0**answers

137 views

### Dimension of global holomorphic sections of a line bundle

Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space ...

**0**

votes

**0**answers

52 views

### Natural correspondence between the set of morphisms and the set of global sections

I'm trying to prove the following claim.
Let $X$ : smooth projective scheme over $\mathbb{C}$, $L$ : line bundle over $X$ and $A$ : $\mathbb{C}$-algebra.
$X_A=X\times_{\mathbb{C}} \operatorname{Spec}A$...