Given a morphism $f: Y \to X$ and a globally generated line bundle $\mathcal L$ on $X$ such that $H^0(X,\mathcal L) \otimes H^0(X,\mathcal L) \to H^0(X,\mathcal L \otimes \mathcal L)$ is surjective.
Which kind of assumptions on $Y,X,f$ and $\mathcal L$ are sufficient to guarantee the same holds for $f^*\mathcal L$, i.e. $H^0(Y,f^*\mathcal L) \otimes H^0(Y,f^*\mathcal L) \to H^0(Y,f^*\mathcal L \otimes f^*\mathcal L)$ is surjective?
I am particularly interested in the case that $X$ is the projective space over a field, $f$ a closed immersion and $\mathcal L = \mathcal O(1)$.
I know that it should be sufficient to require the embedding to be projectively normal in that case. Are there weaker assumptions?
