[Question has been updated with more context and perhaps a better explanation of my question.]
Source: Smith et al., Invitation to Algebraic Geometry, Section 8.4 (pages 131 - 133).
First, a brief set-up, whose purpose will become obvious in a minute. Note that everything here is over $\mathbb{C}$.
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 $X$...
Next, it's shown that this bundle has no global sections, in the case $X = \mathbb{P}^1$:
A global section of the tautological bundle defines, for each point $p \in \mathbb{P}^1$,
a point $(a(p), b(p)) \in \mathbb{C}^2$ lying on the line through the origin corresponding to $p$. Since the assignment $p \mapsto (a(p), b(p))$ must be a morphism, we see that projecting onto either factor, we have morphisms $a, b : \mathbb{P}^1 \rightarrow \mathbb{C}$. But because $\mathbb{P}^1$ admits no nonconstant regular functions, both $a$ and $b$ are constant functions. But then both are zero...
So far, so good. Now:
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.
This, I don't understand. Specifically, whereas the set of $v \in \mathbb{C}^{n+1}$ such that $v \in \ell$ was a subspace of $\mathbb{C}^{n+1}$, the set of linear functionals $f : \ell \rightarrow \mathbb{C}$ appears to be a quotient of $(\mathbb{C}^{n+1})^\ast$, rather than a subspace. So I don't see how to perform the parallel construction here.
In particular, any method that cuts something out of $(\mathbb{C}^{n+1})^\ast \times \mathbb{P}^n$ would seemingly allow us to carry out the argument above and show that the hyperplane bundle has no global sections, which is false.