In Huybrechts' Complex Geometry text, he makes the following claim:
(Claim) "Any homogeneous polynomial $s \in \mathbb{C}[z_o,\dots, z_n]_k$ of degree $k$ defines a linear map $(\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}.$ "
He then writes, using the claim:
"This gives rise to a holomorphic map $\mathbb{P}^n \times (\mathbb{C}^{n+1})^{\otimes k} \to \mathbb{C}$ which is linear on any fibre of the projection to $\mathbb{P}^n$ . The restriction to $\mathcal{O}(-k)$ thus provides a holomorphic section of $\mathcal{O}(k)$. This way we associate to any homogeneous polynomial $s$ of degree $k$ a global holomorphic section of $\mathcal{O}(k),$ which will also be called $s.$"
Is the following map the one he is referring to in his claim?
There is a basis $B_1$ for the space of homogeneous polynomials of degree $k$ which consists in all distinct products of $k-$ tuples of variables. There is a basis $B_2$ for the space $(\mathbb{C}^{n+1})^{\otimes k}$ which consists in all $(n+1)^k$ sequences of basis elements of $C^{n+1}.$ Now $B_2$ is larger than $B_1$, but the two are in 1:1 correspondence modulo reordering of sequence elements in $B_1$. This allows us to define an injective map.