# 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

## 1 Answer

Chuck, I think you can find references for both of the statements 1,2 that Allen gives in Fulton's Intersection theory, Appendix B, Section 5, and maybe also section 7. There, I think (1) is proved more generally for a regularly embedded subscheme. I asked a question about (2) some time ago. You might find what you are looking for (including how one proves 2, which is easy, and "intuition") here:

Tangent bundle and normal bundle in self-product

Fulton doesn't include proofs for everything but gives references to EGA. (Since I don't have the book with me, the section numbers might be a little off, maybe someone can correct me.)

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Great, thanks! That's very helpful. – Chuck Hague Jan 14 '13 at 17:29