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I always confused by whether tautological bundle is $\mathcal{O}(1)$ or $\mathcal{O}(-1)$, and definitions from different sources tangled in my brain. However, I thought this might not be simply a matter of convention.

Let $\mathbb{P}^n$ be a projective space of dimensional $n$, if we realize a point $[l]$ in $\mathbb{P}^n$ as a line $l \subset \mathbb{C}^{n+1}$ passing through origin. Then the tautological bundle $S$ of $\mathbb{P}^n$ is defined as a subbundle of $\mathbb{P}^n\times \mathbb{C}^{n+1}$ by

$$[l] \times l \subset [l] \times \mathbb{C}^{n+1}$$

In many books, the convention is $S \cong \mathcal{O}(-1)$ on $\mathbb{P}^n$. Here $\mathcal{O}(-1)$ is defined as it is in Hartshorne which is the sheaf of modules associated to $\mathbb{C}[x_0,\dots,x_n](-1)$ (for example $x_i^{-1}$ is degree $0$ element in $\mathbb{C}[x_0,\dots,x_n](-1)$). I know something need to be clarified in $S \cong \mathcal{O}(-1)$: for $S$, I mean the sheaf associated to the tautological line bundle S. So it seems the problem becomes to show $S$ does not have global sections (because $\mathcal{O}(-1)$ is different from $\mathcal{O}(1)$ by does not have global sections)?

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2 Answers 2

up vote 11 down vote accepted

As a complement to Matt's fine answer let me explain why $\mathcal O(-1)$ has only zero as global section.

A section $s\in \Gamma(\mathbb P^n,\mathcal O(-1))$ is in particular a section of the trivial bundle $\mathbb P^n \times \mathbb C^{n+1}$, so that it is of the form $s(x)=(x,\sigma (x)) $ with $\sigma:\mathbb P^n \to \mathbb C^{n+1}$ a regular map.
But such a map $\sigma$ is a constant, since any regular map $\mathbb P^n \to \mathbb C$ is constant by completeness of $\mathbb P^n$.
So $\sigma (x)=v\in \mathbb C^{n+1}$, a fixed vector independent of $x$.
However for $x=[l]$, we must have $\sigma (x)=v\in l$.
In other words, that constant vector $v\in \mathbb C^{n+1}$ must lie on all lines $l\subset \mathbb C^{n+1}$, which forces $v=0$ .
We have thus proved that $$\Gamma(\mathbb P^n,\mathcal O(-1))=0$$

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Thank you, it makes a lot sense!! –  Li Yutong Sep 9 '13 at 16:57

The line bundle $\mathcal O(1)$ has as global sections the functions $x_0,\ldots,x_n$ which give coordinates on $\mathbb C^{n+1}$; these are not vectors in $\mathbb C^{n+1}$ but in its dual.

The tautological bundle is as you described, and the elements of its fibres are vectors in $\mathbb C^{n+1}$. Thus its sheaf of sections is dual to $\mathcal O(1)$, and so equals $\mathcal O(-1)$.

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Thank you very much! –  Li Yutong Sep 9 '13 at 16:57

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