Some search on the internet and this site didn't result in any topic about this question of Silverman's The Arithmetic of Elliptic Curves:
Let $W \subset \mathbb{P^n}$ be a smooth algebraic set, each of whose component varieties has dimension $n-1$. Prove that W is a variety.
hint: use Krull's Hauptidealsatz to show W is the zeroset of a single homogenous polynomial.
my ideas: So if $I(V)$ contains at least two linear independent polynomials, we have to prove the height of $I(V)$ is at least $2$.
After that, it is enough to find one singular point, where we assume $I(V)=fg$ with $f,g$ nonconstant polynomials.
If $W$ contains more than one component, then it is the union of projective subvarieties of codimension 1, which necessarily have non-empty intersection. (This is immediate from Bezout but if you prefer a smaller hammer, also follows from the Krull height theorem.) Then, we need to check that $W$ cannot possibly be smooth at a point lying on two or more of these components. We can work affine-locally and assume the point $x=(0,0,\ldots,0)\in\mathbb{A}^n$ lies on multiple components, all of codimension 1; by Hauptidealsatz these components are (possibly after passing to a smaller open subset) cut out by single equations. Hence we need to check if that $f,g$ vanish at $x$, then the hypersurface $V(fg)$ is not smooth at $x$; indeed all of the partials take the form $f'(x)g(x)+f(x)g'(x)=0$ (slightly abusing notation here).
So the conclusion is that $W$ can only have one component, hence it is an irreducible variety.