The homogeneous polynomials in $\mathbb{C}[x_0,\cdots,x_n]$ can be considered as the global sections of a line bundle over $\mathbb{P}^n$ (the line bundle corresponding to Serre's twisting sheaf). In fact if $\mathcal{L}$ is the line bundle corresponding to $\mathcal{O}(1)$ then the homogeneous polynomials in degree $d$ are (isomorphic to) $H^{0}(\mathbb{P}^n, \mathcal{L}^{\otimes d})$. The line bundle $\mathcal{L}$ is globally generated since $\mathbb{P}^n$ can be covered by open sets $U_i = \{x = [x_0:\cdots:x_n] \in \mathbb{P}^n: x_i \not = 0\}$.
According to this example/theorem in Lazarsfeld (2.1.29):
the multiplication map: $H^0(\mathbb{P}^n, \mathcal{L}^{\otimes a}) \otimes H^0(\mathbb{P}^n, \mathcal{L}^{\otimes b}) \rightarrow H^0(\mathbb{P}^n, \mathcal{L}^{\otimes (a+b)})$ is surjective for $a,b$ big enough. This seems wrong to me since it would mean that there are no irreducible homogeneous of arbitrarily high degree, which is not true if $n > 1$.
I'm a beginner in Algebraic Geometry so I would appreciate it if someone can explain to me what is it that I'm missing or misunderstanding about this example. Thanks.