Let $L(-1)$ be the tautological line bundle on $\mathbb{P}^n$. I would like to understand the proof of the following:
Fact. The only global section on $L(−1)$ is the zero section.
Proof. Let $s : \mathbb{P}^n \to L(−1)$ be a holomorphic section. For any $l ∈ \mathbb{P}^n$ we have $s(l)=(l,z_l)$ for some $z_l \in \mathbb{C}^{n+1}$ lying on the line $l$. Thus, $l \to z_l$ is a holomorphic map $\mathbb{P}^n \to \mathbb{C}^{n+1}$. By the maximum principle this map must be constant, so $z_l \equiv w \in \mathbb{C}^{n+1}$. On the other hand $s$ is fiber preserving, so $w \in l$ for each line $l$ through the origin of $\mathbb{C}^{n+1}$. Hence $w = 0$.
How does the maximum principle imply that the map is constant?
Thanks a lot.
EDIT: my question is slightly different from the linked answer as that result is for holomorphic functions (to $\mathbb{C}$), whereas here I have a holomorphic map (to $\mathbb{C}^{n+1}$).