Let $X\subseteq \mathbb{P}^n$ be a projective variety, and let $p\in X$. I define the blow up of $X$ at $p$ as the closure $\Gamma$ in $X\times \mathbb{P}^{n-1}$ of the graph of the projection $\phi$ from $p$.
We have a projection $\pi:\Gamma\to X$. Is it true that $\pi^{-1}(p)=(p, \phi(T_pX))$?
I asked myself this question in order to motivate the fact that the exceptional divisor is often described as $\mathbb{P}(T_pX)$.