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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?

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This theorem does hold over any algebraically closed field; notably we need the fact that the dimension of an affine variety, as given by the length of a maximal chain of irreducible subvarieties, is the same as the krull dimension of its coordinate ring. Here is a nice pdf by Andreas Gathmann on dimension theory: http://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013-c11.pdf .

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.

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