# Proof that every codimension 1 subvariety of $\mathbb{P}^n$ is $V((f))$

I've been told that

Theorem. Every codimension 1 subvariety of $\mathbb{P}^n$ is $V((f))$, where $f$ is some prime homogeneous polynomial.

I'm under the impression that this is true over any algebraically closed field.

However, I'm unable to locate a proof. Does anyone know where a proof of this theorem can be located? A version of this for affine space is here, but a bit light on the details.

Also, in this context, does $V((f))$ mean $V$ of the principal ideal generated by $f$, or is the double bracketing indicative of some kind of a power series construction?

For an irreducible affine variety, the answer to your question is given by remark 11.18. In the projective case, it suffices to notice that the dimension of a space is the same as that of an open dense subset. So intersecting your variety $V$ by a standard open affine (so that the intersection is non empty) will give you a polynomial $f \in k[x_1,\dots,x_n]$ such that $V$ is now given by the vanishing set of the homogenization of $f$.
In regards to $V((f))$, your first interpretation is correct.