# On the blow-up along the diagonal in a product

Let $X$ be a smooth variety and consider the diagonal $\Delta \subseteq X \times X$. It seems to be well-known that the exceptional divisor in the blow-up of $X \times X$ along $\Delta$ is isomorphic to the projectivized tangent bundle $\mathbb P(\mathcal T_X)$ of $X$ but I can't find a reference or a proof; where might I find one?

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This is almost by definition of the algebraic (co)tangent bundle. What is the normal bundle of $\Delta$ in $X\times X$? –  J.C. Ottem Jan 11 '13 at 17:10
You may safely state it in a matter-of-fact tone without any proof: "since the normal bundle to $\Delta$ is isomorphic to $\matcal T_X$" etc. –  Serge Lvovski Jan 11 '13 at 17:49
This is a composite of two standard statements. (1) If $X\subseteq Y$, both smooth, then the exceptional divisor in the blowup of $Y$ along $X$ is the projectivized normal bundle to $X$ inside $Y$. (2) The normal bundle to $\Delta$ inside $X\times X$ is isomorphic to the tangent bundle. As such, I agree with Serge Lvovski. –  Allen Knutson Jan 13 '13 at 2:19