All Questions
Tagged with line-bundles ag.algebraic-geometry
172 questions
9
votes
1
answer
405
views
Conceptual understanding of the Néron–Severi group
I'm trying to understand the importance of the Néron–Severi group $\operatorname{NS}(X)$ when $X$ is, say a complex manifold. My background is in the analytic side so I'm much more familiar with line ...
- 137
1
vote
0
answers
81
views
Is every homogeneous line bundle pulled back from the quotient stack?
Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it ...
- 323
2
votes
0
answers
128
views
Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
1
vote
0
answers
105
views
Isomorphism between spectrum of $\mathcal{O}_{\mathbb{P}^1_{[y_0:y_1]}}[y_0^2t,y_0y_1t,y_1^2t]$ and the line bundle $\mathcal{O}_{\mathbb{P}^1}(2)$
Let $\mathbb{P}^1$ be the projective line over a base field $k$, with homogeneous coordinates $[y_0 : y_1]$. Consider the sheaf of $\mathcal{O}_{\mathbb{P}^1}$-algebras $\mathcal{A} = \mathcal{O}_{\...
- 11
0
votes
1
answer
193
views
Is the square of a special line bundle also special?
Suppose $C$ is a smooth projective curve over, say, $\mathbb{C}$. I'm interested in knowing whether the following is true.
Let $\mathcal{L} \in Pic^d(C)$ be a special line bundle, i.e. its $H^1 \neq 0$...
- 129
4
votes
1
answer
288
views
Characterizing principal polarizations of abelian surfaces
Suppose $X$ is a complex abelian variety of dimension 2. Then I believe the ring of endomorphisms $\mathrm{End}(X)$, tensored with $\mathbb{C}$, is isomorphic to a subalgebra $M_2(\mathbb{C})$ of $2 \...
- 22.3k
1
vote
0
answers
146
views
Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 275
4
votes
1
answer
333
views
Question regarding the definition of linearization of line bundles
I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...
- 737
1
vote
1
answer
141
views
Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors?
Oguiso writes[1]
Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= ...
- 1,286
7
votes
0
answers
242
views
Mumford's definition of an abelian variety's $Pic^0$
I'm not sure whether this is a research-level question, but upon skimming through Mumford book of Abelian Varieties I noticed he gives this definition
$$
\begin{equation}
\label{eq}
\text{Pic}^0(A)=\{\...
- 71
3
votes
1
answer
248
views
A question on "Ample subvarieties of algebraic varieties"
Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety ...
- 8,998
1
vote
0
answers
277
views
How to define Cartier divisor and Weil divisor on algebraic stack?
How to define Cartier divisor and Weil divisor on algebraic stack? Do they correspond to line bundles on stack like the case of schemes? In case of a Deligne-Mumford stack, can we have a simpler ...
- 523
3
votes
0
answers
208
views
Metric on line bundle defined fiberwise
Let $(X,\mathcal{O}_X)$ be an analytic space, and let $L$ be a line bundle on $X$. Intuitively, a metric $||\cdot||$ is a continuous choice of a metric for each fiber of the line bundle, which is a ...
- 131
4
votes
1
answer
362
views
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
3
votes
0
answers
218
views
Line bundles with meromorphic transition functions
I have the following situation: let $X$ be a projective complex manifold and let $f \in H^1(X,\mathcal{M}^{\times})$. So $f$ defines something like a line bundle with meromorphic transition functions.
...
- 153
1
vote
0
answers
330
views
Computing Picard groups of arbitrary quadric hyperplane
I know the Picard group of a smooth two dimensional quadric surface is $\mathbb Z^2$, but I am wondering if the computation can be generalized to higher dimension? In particular, is the Picard group ...
- 231
3
votes
1
answer
493
views
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 ...
- 18.4k
2
votes
1
answer
193
views
Linear system giving the projective embedding of the tangential variety
I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...
- 1,343
6
votes
1
answer
340
views
Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor
Let $X$ be a complex compact manifold, and write $\mathcal{O}_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}_X^{\times}$ be the subsheaf consisting of holomorphic functions. These ...
- 5,966
1
vote
0
answers
200
views
Bundles vs. line bundles
Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...
user30211
2
votes
0
answers
242
views
Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
- 8,998
1
vote
1
answer
190
views
Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve
We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
- 11
5
votes
1
answer
287
views
Extension of first order deformations of a line bundle
Let $X$ be a smooth complex algebraic variety with $H^0(X,\mathcal{O}_X) = \mathbb{C}$ and $V \subset X$ an open subvariety whose complement has codimension two. Now, let $L_{\varepsilon}$ be a line ...
3
votes
0
answers
265
views
Ample divisor of degree two on a blow-up of $\mathbb P^2$ at nine points
Let $\pi:S \rightarrow \mathbb P^2$ be a blow-up at nine points in general position.
I am finding an ample divisor $L$ on $S$ of degree two ($L^2=2$).
Since $Pic(S) = \mathbb Z h \oplus \mathbb Z e_1 \...
- 1,841
1
vote
0
answers
151
views
Extension of a section of a line bundle on a family of curves to the central fibre
I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line ...
- 341
2
votes
0
answers
92
views
Linear system of a relative effective divisor on an arithmetic surface contains vertical divisors
I am puzzled by the behavior of some divisors in my attempt to understand the relative Picard functor $\mathrm{Pic}_{X/S}$ of an arithmetic surface $\pi:X\to S$. This is defined by relative divisors $...
- 1,142
2
votes
1
answer
204
views
Morphism attached to a big and globally generated line bundle
Let $X$ be a smooth projective variety over a field $k$. Let $L$ be an invertible sheaf on $X$. Suppose that $L$ is big and globally generated. Can one conclude that the associated morphism $\phi_L : ...
- 63
5
votes
1
answer
390
views
Compact complex non-Kähler manifolds with nef canonical bundle
Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?
Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...
- 275
4
votes
0
answers
1k
views
Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?
Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
- 4,164
3
votes
1
answer
268
views
When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?
This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question.
Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
- 1,338
4
votes
1
answer
362
views
Type vs degree of a polarized abelian variety
Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that
$d = \chi(L) = \dim H^0(A,L)$
since $L$ is ample.
I've read in a lot ...
5
votes
0
answers
321
views
Non-trivial line bundle on $\mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$
A line bundle is a holomorphic complex-dimension-one bundle on a complex manifold.
The complex manifold $X = \mathbb{C}^{\ast} \times \mathbb{C}^{\ast}$ admits a non-trivial line bundle for the ...
- 103
3
votes
1
answer
339
views
Galois invariant line bundle and base change
Let $K$ be a number field and consider a finite Galois extension $L|K$. Moreover let $X$ be a projective, regular, integral variety over $K$. After a base change we obtain a morphism of varieties $f:...
- 321
1
vote
1
answer
483
views
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
1
vote
0
answers
255
views
Construction of the Hilbert Scheme
I am reading the book "Rational Curves on Algebraic Varieties" of János Kollár. Definition-Proposition 1.2, begin like this:
Let $g:Y\rightarrow Z$ be a projective morphism and $\mathcal{O}(...
- 519
1
vote
1
answer
303
views
(Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?
I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper,
https://arxiv.org/abs/math/0411613.
However I cannot purpose analogously on $D^b(P^1 \times P^2)$.
...
- 63
1
vote
0
answers
255
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 ...
- 55
0
votes
0
answers
120
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{...
- 297
1
vote
0
answers
469
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 ...
- 35
1
vote
0
answers
99
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,
...
- 297
2
votes
0
answers
152
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$ ...
- 1,066
1
vote
0
answers
124
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)...
- 297
2
votes
0
answers
677
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}_{...
- 1,343
7
votes
1
answer
506
views
Multiplication maps for big line bundles
In Birational Geometry of Algebraic Varieties, Kollar and Mori write that for a line bundle "being big is essentially the birational version of being ample" (page 67). Recall that a line ...
5
votes
1
answer
443
views
Line bundle on product scheme
Let $k$ be a field, $X$ be a complete variety over $k$, $V$ be an open subvariety of $X$, $Y$ be a scheme over $k$. Suppose $L$ is a line bundle on $V\times Y$. If $L|_{V\times\lbrace y\rbrace}$ ...
- 51
3
votes
1
answer
168
views
Weights on the linearization
Consider, just as an example, an action of $\mathbb{C}^*$ on $\mathbb{P}^2$ of the form
$$t\cdot p=[p_0:tp_1:t^2p_2]$$
There are $3$ fixed points, namely $e_1,e_2,e_3$. If I consider a $\mathbb{C}^*$-...
- 133
3
votes
1
answer
529
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....
- 828
1
vote
0
answers
182
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
0
answers
229
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 ...
- 14.2k
3
votes
1
answer
276
views
Polarization of an abelian variety made by the sum of two divisors
Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$.
In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...