Suppose $X$ is a variety (irreducible quasi-projective algebraic set) of dimension $d$, $x\in X$, and $\pi:\tilde X\to X$ the blow-up of $X$ at $x$. Then is it true in general that the exceptional divisor of the blow-up is isomorphic to projective space of dimension $d-1$? Is it at least true for smooth projective surfaces over $\mathbb C$?
Beauville's Complex Algebraic Surfaces claims it is, but I do not understand the construction there, so I can't verify. My question is about a proof using the usual construction of the blow-up as the graph of the projection $X\dashrightarrow\mathbb P^{n-1}$ (where $X\subseteq\mathbb P^n$).