Chapter 1, Exercise 2.8: Hypersurfaces and dimension $n-1$

A projective variety $Y\subseteq\mathbb P^n$ has dimension $n-1$ if and only if it is the zero set of a single irreducible homogeneous polynomial $f$ of positive degree. $Y$ is called a hypersurface in $\mathbb P^n$.

Let $Y=Z(f)$ be the zero set of one irreducible polynomial with $\deg f>0$. This has projective coordinate ring $S(Y)=S/(f)$. By the Hauptidealsatz $\operatorname{height}(f)=1$ (using $\deg f\neq 0$) and thus $\dim S(Y) = \dim S - \operatorname{height}(f)=n+1-1=n$. But $\dim Y=\dim S(Y)-1=n-1$.

Now suppose $\dim Y=n-1$, then $\dim S(Y)=n$ and further $I(Y)$ is prime. Also, $\operatorname{height}I(Y)=\dim S(Y)-\dim Y = 1$, so $I(Y)$ has a single nonconstant generator $g$ which can be taken to be homogeneous since $I(Y)$ is a homogeneous ideal. Then $I(Y)=(g)$ implies $Y=Z(g)$ as desired.