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.

Filter by
Sorted by
Tagged with
5 votes
0 answers
222 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 ...
Stefano's user avatar
  • 625
2 votes
2 answers
840 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 $X$ ...
Puzzled's user avatar
  • 8,842
3 votes
2 answers
865 views

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 ...
Puzzled's user avatar
  • 8,842
4 votes
1 answer
298 views

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 ...
Armando j18eos's user avatar
2 votes
0 answers
333 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 ...
user avatar
1 vote
1 answer
224 views

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 $X$?
user avatar
6 votes
1 answer
498 views

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 ...
George's user avatar
  • 113
4 votes
0 answers
171 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 ...
Stefano's user avatar
  • 625
1 vote
0 answers
67 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 ...
i87456's user avatar
  • 141
1 vote
0 answers
280 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 ...
i87456's user avatar
  • 141
3 votes
2 answers
597 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....
Irfan Kadikoylu's user avatar
1 vote
1 answer
244 views

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 $D_0$ ...
Stefano's user avatar
  • 625
2 votes
0 answers
185 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. ...
Drew's user avatar
  • 1,469
1 vote
1 answer
1k views

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 ...
user avatar
1 vote
1 answer
233 views

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 $...
user avatar
0 votes
1 answer
188 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 '...
danielg's user avatar
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 ...
Alex M.'s user avatar
  • 5,207
2 votes
0 answers
230 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 ...
Meer Ashwinkumar's user avatar
3 votes
1 answer
360 views

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 ...
Bernie's user avatar
  • 1,015
2 votes
1 answer
475 views

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\...
user avatar
4 votes
0 answers
168 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 ...
Han Jin Ma's user avatar
2 votes
1 answer
300 views

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 ...
Alan Muniz's user avatar
8 votes
1 answer
561 views

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|...
Qfwfq's user avatar
  • 22.7k
1 vote
0 answers
169 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 ...
Mathemage's user avatar
4 votes
1 answer
841 views

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^*\...
Nicholas Proudfoot's user avatar
2 votes
0 answers
159 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 $n>0$, ...
Jana's user avatar
  • 2,022
1 vote
0 answers
148 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 ...
Irfan Kadikoylu's user avatar
2 votes
2 answers
756 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 ...
Zhaoting Wei's user avatar
  • 8,657
7 votes
2 answers
592 views

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 ...
Drew's user avatar
  • 1,469
2 votes
1 answer
368 views

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 ...
Drew's user avatar
  • 1,469
0 votes
1 answer
1k views

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?
John Robertson's user avatar
2 votes
3 answers
2k views

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?
user avatar
3 votes
4 answers
3k views

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(...
user avatar
2 votes
1 answer
410 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 ...
DonD's user avatar
  • 251
0 votes
0 answers
176 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 ...
Ritwik's user avatar
  • 3,235
2 votes
0 answers
231 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 ...
Drew's user avatar
  • 1,469
2 votes
2 answers
507 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 ...
Christian Fischmann's user avatar
1 vote
1 answer
183 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 $r>1$....
Kali's user avatar
  • 503
20 votes
1 answer
2k views

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?...
Dan Petersen's user avatar
  • 39.2k
0 votes
1 answer
239 views

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 ...
user avatar
3 votes
2 answers
5k views

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 ...
user avatar
0 votes
1 answer
136 views

Two questions about line bundles over Kuranishi families

i'm studying the article "Variétés Kahleriennes dont la première classe de chern est nulle" by Arnaud Beauville and i have a couple of questions i would like to ask you, hoping they are not too ...
Paz Deal's user avatar
2 votes
1 answer
179 views

When is the Clifford index of a curve computed by pencils?

Under which circumstances is the Clifford index of a curve computed by pencils?
Heitor's user avatar
  • 761
0 votes
2 answers
282 views

Restriction of a line bundle to a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...
Ali Shehper's user avatar
0 votes
1 answer
342 views

Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
user avatar
1 vote
1 answer
123 views

Unicity of a vector field on $S^1$-bundle

Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...
user avatar
0 votes
4 answers
451 views

An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to $$A=\{f:L^{\times}\to \mathbb{...
user avatar
0 votes
2 answers
427 views

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 ...
Appliqué's user avatar
  • 1,269
2 votes
2 answers
1k views

Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
user avatar
1 vote
0 answers
251 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 ...
user42066's user avatar