Show that any non-trivial homogeneous polynomial $0 \ne s \in \mathbb{C}[z_0, \dots, z_n]$ of degree $k$ can be considered as a non-trivial section of $\mathcal{O}(k)$ on $\mathbb{P}^n$.
So $\mathcal{O}(k) = \mathcal{O}(1)^{\otimes k} = (\mathcal{O}(-1)^*)^{\otimes k}$ and we want to consider $s$ as a holomorphic map $$s:\mathbb{P}^n \to \mathcal{O}(-1)^* \otimes \dots \mathcal{O}(-1)^*.$$
Now since $s$ is homogeneous we also have that $$\lambda^k s(z_0, \dots, z_n) = s(\lambda z_0, \dots, \lambda z_n).$$
My issue is not understanding $\mathcal{O}(-1)^*$ properly and thus I can't see how $s$ is a map from the projective space to the tensor product. I have seen that the fiber of $$\pi: \mathcal{O}(-1)^* \to \mathbb{P}^n$$ is isomorphic to $\ell^*$ if we consider the elements of $\mathcal{O}(-1)$ as pairs $(\ell, z)$ with $z \in \ell$. Can we use only the information about the fibers to show that $s$ is this kind of a map?