# Line bundles, line bundles on a homogeneous space, and sections of line bundles

I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and sections of line bundles. These concepts are on page 140 (the first paragraph of section 9.3) of the book Young Tableaux by Fulton.

I know that $R \times S^{1}$, where $S^{1}$ is the unit circle and $R$ is the real line, is a line bundle. Möbius strip is also a line bundle. Are there other specific examples of line bundles?

How to understand the sentence in the first paragraph of section 9.3 (page 140) in the book of Fulton clearly: There is a general procedure for producing representations as sections of a line bundle on a homogeneous space?

I also have a question in the first paragraph of the book Young Tableaux on page 131. It is said that if $F$ is a subspace of $E$ of co-dimension $d$, then the kernel of the map from $\wedge^{d}(E)$ to $\wedge^{d}(E/F)$ is a hyperplane in $\wedge^{d}(E)$; assigning this hyperplane to $F$ gives a mapping $Gr^{d}E \to P^{*}(\wedge^{d}E)$. What are the maps $\wedge^{d}(E) \to \wedge^{d}(E/F)$ and $Gr^{d}E \to P^{*}(\wedge^{d}E)$ explicitly?

Thank you very much.

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Step 1 in learning about line bundles: learn to spell them correctly. – Pete L. Clark Jan 4 '11 at 20:09
Hi Pete, thank you. – user Jan 4 '11 at 20:15

## 1 Answer

The formation of exterior powers is a functor, so the quotient map $E \to E/F$ induces a map $\wedge^d E \to \wedge^d(E/F)$. It is not hard to see that this latter map is surjective. Since $E/F$ is of dimension $d$ by assumption, its top exterior power $\wedge^d(E/F)$ is one-dimensional, and so the kernel of $\wedge^d E \to \wedge^d(E/F)$ is of codimension one, i.e. is a hyperplane.

Now given $F$, this kernel is a hyperplane in $\wedge^d E,$ and so we get a map $Gr^dE \to P^*(\wedge^d E)$ sending each element $F$ of the source (i.e. each codimension $d$ subspace $F$ of $E$) to the kernel of the associated map $\wedge^d E \to \wedge^d (E/F)$, which as we have just seen is a hyperplane in $\wedge^d E$, i.e. is an element of $P^*(\wedge^d E)$.

As for your question about line bundles, you might want to learn more basic theory before trying to understand the statement about sections of homogeneous line bundles. For example, the examples you wrote down are line bundles in the smooth category, whereas Fulton is talking about line bundles, and sections thereof, in the algebraic category.

One place to start would be by learning about the tautological bundle over a projective space, since this is an example of the kind of homogeneous line bundle that Fulton is talking about.

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Hi Matt, thank you very much. Which book could I read to learn the tautological bundle? – user Jan 4 '11 at 23:56
@Jianrong: Dear Jianrong, The beginning of Milnor and Staffesh discusses projective spaces and the tautological bundles over them, from a smooth manifold point of view. These also exist as algebraic line bundles over projective space thought of as an algebraic variety. The sheaf of sections of the tautological bundles is then denoted $\mathcal O(-1)$. If you read Hartshorne's discussion of projective space, you will find plenty about $\mathcal O(1)$ (which is the dual to $\mathcal O(-1)$, and is actually more natural from the algebraic geometry view-point), but Hartshorne doesn't make ... – Matt E Jan 5 '11 at 16:00
... the connection between the sheaves $\mathcal O(-1)$ and $\mathcal O(1)$ and the underlying line bundles very explicit. (His text focusses very much on the sheaf point of view, and the connection with the line bundles or vector bundles which underly the locally free sheaves that he talks about tends to be left to the exercises). If you want more answers (and I'm sure that there are people here who know good references for this) you could ask another question about this directly, something like: what are good references for learning about vector bundles and their sheaves of sections ... – Matt E Jan 5 '11 at 16:05
..., mentioning that you are most interested in the algebraic geometry context, and making it explicit that you are a beginner, so that references giving clear explanations of things like the tautogolical line bundle on $\mathbb P^n$, its dual, and the associated sheaves of sections $\mathcal O(-1)$ and $\mathcal O(1)$ would be especially welcome. – Matt E Jan 5 '11 at 16:07
Hi Matt, thank you very much. – user Jan 5 '11 at 18:25