Source: Smith et al., Invitation to Algebraic Geometry, Section 8.4 (pages 131 – 133).
I have a very limited background in algebraic geometry and I'm trying to understand line bundles. In the book they define the tautological bundle as follows:
The tautological bundle over $\mathbb{P}^n$ is constructed as follows. Consider the incidence correspondence of points in $\mathbb{C}^{n+1}$ lying on lines through the origin, $B = \{(x, \ell) \;|\; x \in \ell \} \subseteq \mathbb{C}^{n+1} \times \mathbb{P}^n$, together with the natural projection $\pi : B \rightarrow \mathbb{P}^n$. […] The tautological bundle over the projective variety $X \subseteq \mathbb{P}^n$ is obtained by simply restricting the correspondence to the points of $X$…
Later on in the text they define the hyperplane bundle as
The hyperplane bundle $H$ on a quasi-projective variety is defined to be the dual of the tautological line bundle: The fiber $\pi^{-1}(p)$ over a point $p \in X \subset \mathbb{P}^n$ is the (one-dimensional) vector space of linear functionals on the line $\ell \subset \mathbb{C}^{n+1}$ that determines $p$ in $\mathbb{P}^n$. The formal construction of $H$ as a subvariety of $(\mathbb{C}^{n+1})^\ast \times \mathbb{P}^n$ parallels that of the tautological line bundle.
I can't see how to put a variety structure on the line bundle?
Best Answer
The construction is exactly the same, taking the couples $(\phi,p)$ where $p \in \Bbb P(V)$ and $\phi : p \to \Bbb C$ is a linear functional.
If you know that line bundle are determined by cocycles, then compute the cocycles of $\mathcal O(-1)$ ( this is a notation for tautlogical line bundle), and the inverse cocycles will be the cocycles of H, this will gives you all the information you need about $H$.
Remark : this bundle is named like this because any $\mu \in V^* \backslash \{0\}$ create a section of our bundle, namely $\sigma : \Bbb P(V) \to H, L \mapsto \mu_{|L} $. Notice that the kernel of $\sigma$ is an hyperplane, hence the name. On the other hand, there is no algebraic regular section $\sigma : \Bbb P^n \to \mathcal O(-1)$.