2
votes
1answer
31 views

Sheaf of sections vanishing at a point is $\Gamma(E) \otimes I_p$

Notation: $E$ is a vector bundle with sheaf of sections $\Gamma(E)$. $I_p$ is the sheaf of regular functions on the base space vanishing at a point $p$. $\Gamma_p(E)$ is the sheaf of sections of $E$ ...
3
votes
0answers
45 views

Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
0
votes
1answer
54 views

Quotient of locally free sheaf is locally free?

If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ ...
1
vote
1answer
52 views

Group of Automorphism of the Fiber of Fiber Bundle

Let us consider the Mobius Bundle. Can Someone Explain the part "There is no natural unique homeomorphism of $Y_x$with $Y$. However there are two such which differ by the map $g$ of $Y$ on itself ...
4
votes
1answer
53 views

Lifting vector bundles along thickenings

Let $X_0 \to X$ be a nilpotent closed immersion of schemes. Is every vector bundle on $X_0$ the pullback of a vector bundle on $X$? The answer is yes when $X$ is affine. In general, there may be ...
1
vote
1answer
27 views

Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
1
vote
1answer
51 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
3
votes
1answer
83 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
0
votes
0answers
28 views

Just a definition of an algebraic bundle, bundles on $\mathbb{P}_n$

I have just realized the notion of an algebraic vector bundle. I have some questions. In particular, I'd like to understand wheather I understand it correctly. Let $$\pi:E\longrightarrow X$$ be a ...
4
votes
0answers
56 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
1
vote
2answers
61 views

An extension of line bundles splits locally

Consider an extension $0\rightarrow L \overset{\alpha}{\rightarrow} E \overset{\beta}{\rightarrow} L' \rightarrow 0$ of bundles and bundle homomorphisms, where $L$ and $L'$ are line bundles. (Let's ...
0
votes
1answer
36 views

The pullback of a nontrivial line bundle is nontrivial?

Let $X$ be a complex manifold and $E$ a holomorphic vector bundle over $X$ of rank $2$. Let $\mathbb{P}(E)$ denote the projectivization of $E$, with the natural map $p: \mathbb{P}(E) \rightarrow X$. ...
2
votes
0answers
65 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
1
vote
0answers
33 views

Homotopic maps induce isomorphic pullbacks (algebraic setting)

I know that in continuous or even smooth category, if $f_i: X \to Y$ is homotopy and $B \to Y$ is vector bundle, than $f_0^* (B) \cong f_1^* (B)$. But I wonder whether there is the same in algebraic ...
0
votes
0answers
63 views

Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
6
votes
1answer
64 views

exact sequence induced by restriction to closed subscheme

I'm somewhat embarrassed to be confused about this issue after a while studying algebraic geometry, but here goes. Let $\iota: Y \hookrightarrow X$ be the inclusion of a closed subscheme, $U$ is the ...
4
votes
1answer
107 views

What's the intuition for the fact that $\mathscr{O}(-k)$ and $\mathscr{O}(k)$ are so different?

maybe this question makes no sense and I just cannot accept the fact that dual the line bundle is different from the respective line bundle itself. Since it looks like that manifolds are more ...
3
votes
1answer
116 views

Exactness of the pullback of the Euler sequence

Let $(L,V)$ be a base point free $g^r_d$ on a curve $C$. Then we have an exact sequence $0\rightarrow M_{L,V}\rightarrow\mathcal{O}_C^{r+1}\xrightarrow{\theta} L\rightarrow 0$, where we just let ...
4
votes
1answer
79 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
1
vote
0answers
41 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
1
vote
0answers
38 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
1
vote
0answers
59 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
3
votes
1answer
38 views

Boundedness of the operator $[\Lambda, \Theta]$.

I am reading Griffiths-Harris book on algebraic geometry and in "Theorem B", where he proves (the analogue of Serre's vanishing theorem) that Let $M$ be a compact, complex manifold, $L\rightarrow ...
1
vote
0answers
36 views

Vector bundle on a transversal intersection

Consider a curve $X$ (over a field $k$), which is an union of two smooth and connected curves $C$ and $D$ meeting at exactly one point $P \in X$ transversally. Let $E$ be a vector bundle on $X$ such ...
3
votes
0answers
33 views

Total space of a geometric vector bundle

Suppose that $X$ is a scheme with a $B$-action, and that $\lambda: B \to \mathbb{C}^*$ is a character of $B$. We then have a geometric vector bundle $\pi: X \times^B k \to X/B$, where $B$ acts on $X ...
2
votes
0answers
130 views

Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
4
votes
1answer
58 views

Krull-Schmidt-Remak for vector bundles

I'm reading Nori's paper The fundamental group scheme, and I have some problems in certain passages of the proofs. This one is from chapter 1, 2.3. Let $X$ be a complete connected reduced ...
4
votes
1answer
52 views

Ampleness, Nakai's criterion and pullback

In the book I'm reading ( Geometry of Algebraic Curves ), at some point (page $310$) they make the following claim: One can use Nakai's criterion to establish the general fact that if $f:X\to Y$ ...
2
votes
1answer
106 views

Normal bundle of a hyperplane section

Let $Y\subset \mathbb{P}^n$ be a smooth projective variety and let $H$ be a smooth hypersurface in $\mathbb{P}^n$ such that $Z=Y\cap H$ is smooth. How are the normal bundles of the various embeddings ...
3
votes
2answers
83 views

Why is this line bundle clearly ample?

Reading a book I encountered the following claim, which I don't understand. Let $X$ be a smooth projective curve over $\mathbb{C}$, and $q\in X$ a rational point. Denote by $\pi_i: X^n\to X$ the ...
0
votes
1answer
36 views

Equality for rigidified line bundles

Let $L$ and $M$ be two rigidified line bundles (see below for the definition) over a scheme $X\to S$, and assume we know that $ L \cong M\otimes F $ for some line bundle $F$. $$\text{Is it true ...
0
votes
2answers
67 views

How to use Nakayama's lemma here?

Let $X$ and $Y$ be algebraic varieties, with $X$ complete, and let $\pi: X\times Y \to Y$ be the projection onto $Y$. Let $L$ be a line bundle over $X\times Y$ and consider the natural map $$ \alpha: ...
2
votes
0answers
103 views

Simple Examples of Correspondence Between Line Bundles and Divisors

I'm trying to work out some simple examples demonstrating the correspondence between line bundles and divisors. The toric variety $X = \mathrm{Bl}\,_0(\mathbb{C}^2/\mathbb{Z}_2)$ has quotient ...
3
votes
1answer
64 views

Line bundles pullback trick

I'm trying to prove a statement about line bundles, and the following question is crucial to complete a possible proof that I have in mind. Could you give me any hint please? Let $A$ and $B$ two ...
2
votes
1answer
78 views

a question about space of smooth sections

Let $\Gamma(M,L) $ be the space of smooth sections, then why $\Gamma(M,L) $ is isomorphic to $A=\{f:L^{\times}\to \mathbb{C}; f(cz)=c^{-1}f(z), c\in \mathbb{C}-\{0\} , z\in L^{\times}\}$ . Here ...
3
votes
1answer
92 views

The evaluation map for a skyscraper sheaf on an elliptic curve

Let $E$ be an elliptic curve over a field, $z \in E$ is a point, and $d \geq 1$. I consider a skyscraper sheaf $\mathcal{O}_z/m_z^d$, the evaluation map $$ \operatorname{Hom}(\mathcal{O}, ...
2
votes
0answers
40 views

Why can the group of isomorphism classes of line bundles be identified with $H^1(C,\mathbb O_C^*)$?

This is a reference request to the fact in the title. Is there a book at most as advanced as Hartshorn's which explains this result?
4
votes
1answer
176 views

Why does the degree of a line bundle equal the degree of the induced map times the degree of the image plus the degree of the base locus?

Let $L$ be a line bundle on a smooth curve $C$. If $L$ is rank $r+1$, define the induced map (as Arbarello, Cornalba, Griffiths, Harris): $$\begin{aligned}\phi :& C \rightarrow \mathbb ...
3
votes
1answer
39 views

Why is it better to have the induced map by a line bundle $L$ into projective space map into $\mathbb P |L|^*$?

Let $L$ be a line bundle on a smooth curve $C$. If $L$ is rank $r+1$, we get a induced map $C \rightarrow \mathbb P^r$. Why do many sources (as Arbarello, Cornalba, Griffiths, Harris), define this map ...
2
votes
1answer
39 views

Why is $H^0(C,\mathcal O(D))$ a vector space?

Given a divisor $D$ on a smooth curve, one can define the sheaf $\mathcal O(D)$ by the prescription $\Gamma(U,\mathcal O(D) :=$ $\{$meromorphic functions on $U$ that satisfy $(f) + D \ge 0\}$. Then, ...
3
votes
1answer
225 views

Determinant of a tensor product of two vector bundles

Let $X$ be a smooth variety over a field, $V_1$ and $V_2$ are two vector bundles over $X$ of ranks $r_1$ and $r_2$ respectively. Determinant of a vector bundle is the top exterior power of the vector ...
2
votes
1answer
48 views

Quick question: sufficiently positive divisor

I am referring to line 1-2 of page 472 in Griffiths Harris. M is a compact complex manifold of dimension 2 that may be embedded in projective space. L is an arbitrary line bundle on M. How can one ...
2
votes
1answer
115 views

Classification(s) of vector bundles

Algebraic isomorphism classes of vector bundles of rank $r$ on $\mathbb P^1_\mathbb C$ are in bijective correspondence with $r$-tuples of integers $a=(a_1,\dots,a_r)$ such that $a_1\geq a_2\geq \dots ...
2
votes
0answers
82 views

How can I get 3264 conics from chern class?

I'm studying Algebraic geometry by "Enumerative Geometry And String Theory" [Katz]. In section 8.3, he computed the excess contribution 31 and concluded that the number of smooth conics tangent to ...
8
votes
1answer
142 views

Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
2
votes
1answer
115 views

Algebraic line bundles over $\Bbb{P}^1$ : Why is it enough to assume trivialization is given over the standard affine cover?

I am reading this MO post here about the classification theorem of vector bundles over $\Bbb{P}^1$. However, I am mainly interested in the case of just line bundles. Now if the general definition of ...
3
votes
0answers
94 views

Triviality of (equivariant) holomorphic vector bundles

Let $G$ the 1-dimensional diagonalisable linear complex analytic group $\mathbb C^*$. We suppose that $G$ acts linearly on $\mathbb C^n$ with positive weights. Set $X=\mathbb C^n -\lbrace 0 \rbrace$. ...
8
votes
2answers
222 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
4
votes
1answer
103 views

Identifying a line bundle on $\mathbb{P}^1$

I have a geometric line bundle $L$ on $\mathbb{P}^1 = \{[x_0:x_1]\}$. With respect to the standard affine cover $U_0 = \{x_0 \neq 0\}$ and $U_1 = \{x_1 \neq 0\}$, I have the transition function ...
3
votes
0answers
73 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...