Intereting Posts

Entire function with vanishing derivatives?
Is the series $\sum_{n=2}^\infty\frac{(-1)^{\lfloor\sqrt{n}\rfloor}}{\ln{n}}$ convergent?
Study of a function on interval $$
How to show that $\sum_{n=1}^{\infty} \frac{1}{(2n+1)(2n+2)(2n+3)}=\ln(2)-1/2$?
Prove quotient is a perfect square
Extension of Riemannian Metric to Higher Forms
Prove the triangle inequality involving complex numbers.
Exponential map of Beltrami-Klein model of hyperbolic geometry
Every group with 5 elements is an abelian group
Homology groups of the Klein bottle
Verification of integral over $\exp(\cos x + \sin x)$
proof Intermediate Value Theorem
Lagrange method with inequality constraints
Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$.
The Affine Property of Connections on Vector Bundles

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

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

- cohomology of the pullback of a sheaf
- Where to learn algebraic analysis
- Relationship between very ample divisors and hyperplane sections
- Does a regular function on an affine variety lie in the coordinate ring?(Lemma 2.1, Joe Harris)
- Algebraic versus topological line bundles
- closed immersion onto an affine scheme - showing affineness

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)?

- Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?
- Doing Complex Analysis on the Riemann Sphere?
- Surjection from a Noetherian ring induces open map on spectra?
- pullback of rational normal curve under Segre map
- Number of points on an elliptic curve over $ \mathbb{F}_{q} $.
- Are there online English proofs of preservation of properties of morphisms by fpqc descent?
- Does a non-empty locally closed subset of a $k$-scheme of finite type always contain a closed point?
- What is the meaning of normalization of varieties in complex geometry?
- Spectrum of $\mathbb{Z}^\mathbb{N}$
- Singular and Sheaf Cohomology

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 $$\Gamma(\mathbb P^n,\mathcal O(-1))=0$$

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 $\mathcal O(-1)$.

- Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?
- Finite additivity in outer measure
- Finding Rational numbers
- showing that $\lim_{x\to b^-} f(x)$ exists.
- Determinant of a special $0$-$1$ matrix
- radical expression
- When does a system of equations have no solution?
- If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
- How to prove that nth differences of a sequence of nth powers would be a sequence of n!
- Finding $\sum\limits_{n=2}^\infty \ln\left(1-\frac{1}{n^2}\right)$
- Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$
- Sum of averages vs average of sums
- The Continuum Hypothesis & The Axiom of Choice
- Why does Newton's method work?
- Prove that the limit exists of an increasing and bounded function