If a subscheme of codimenion one in $\mathbb{P}^n_k$ is define by ideal sheaf $\mathcal{I}$, is the saturation $\oplus_{n\ge 0}\Gamma(I(n))\subseteq k[x_0,...,x_n]$ be generated by one element?
Is there a counterexample? Do we require the subscheme to be locally principal? (Is there a closed subschemem of projective space which is not locally principal?)