I am studying the paper of Matsumura & Monsky on the automorphisms of hypersurfaces and I can't understand a part of the proof of theorem 2. I restate it for clarity.
For $d \geq 5$ let $H_d$ be a non-singular hypersurface of degree $d$ in $\mathbb{P}^3$. We want to prove that Aut$(H_d)$=Lin($H_d)$ where Aut$(H_d)$ is the group of automorphisms and Lin$(H_d)$ is its subgroup consisting of elements induced by the projective transformations of $\mathbb{P}^3$ which leave $H_d$ invariant. We take a linear system of hyperplane sections $L_1$ on $H_d$. It is stated that $L_1$ is projectively normal, and thus complete.
I don't understand how we can directly say that $L_1$ is projectively normal, I know that it is true for any curve in $\mathbb{P}^2$ but I don't know a proof for it in this case.