We have shown that for an affine cone of a nonempty subset $X$ of $\mathbb P^n$, i.e $C(X)$, that the irreducible decomposition of $C(X)$ given by $Y_1 \cup \dots \cup Y_m$ has that $Y_i = C(X_i)$ for some closed $X_i \subseteq \mathbb P^n$.
I wanted to know, what exactly is $X_i$ in this case? I was perhaps thinking that the $X_i$ would be the irreducible components of $X$, but was curious on if this is true and how one may prove it.
