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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1$\begingroup$ This is almost by definition of the algebraic (co)tangent bundle. What is the normal bundle of $\Delta$ in $X\times X$? $\endgroup$– J.C. OttemCommented Jan 11, 2013 at 17:10
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2$\begingroup$ 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. $\endgroup$– Serge LvovskiCommented Jan 11, 2013 at 17:49
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2$\begingroup$ 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. $\endgroup$– Allen KnutsonCommented Jan 13, 2013 at 2:19
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1$\begingroup$ mathoverflow.net/questions/111430/… $\endgroup$– Sándor KovácsCommented Jan 13, 2013 at 22:57
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$\begingroup$ Sorry, but I am a bit of confused, by [Hartshorne] II Thm. 8.24, the exceptional divisor in the blowing-up of $Y$ along $X$ is isomorphic to $\mathbf P(\mathcal C_{Y/X})$, so isn't $\mathbf P(\Omega_X)$ instead of $\mathbf P(\mathcal T_X)$? $\endgroup$– Aoi KoshigayaCommented Aug 5, 2021 at 7:55
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1 Answer
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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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$\begingroup$ Great, thanks! That's very helpful. $\endgroup$ Commented Jan 14, 2013 at 17:29