Let $j:X\subset \mathbb P_{\mathbb C}^n$ ($n\geq 3$) be a hypersurface, defined by a section of a very ample line bundle $\mathcal L$, with a ordinary double point $P$ as the only singularity and $\tau:\tilde X\rightarrow X$ the blow-up of the double point. Fulton (4.2.6) define in the Grothendieck group of vector bundle, a virtual tangent bundle for $X$. Is it true that, in the Grothendieck group of $\tilde X$, we have the equality $$[T_{\tilde X}]=\tau^*j^*([T_{\mathbb P^n}] - \mathcal L) - i_*[T_{E/P}(E)]$$ where $i_*:E\rightarrow \tilde X$ is the inclusion of the exceptional divisor?
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2$\begingroup$ Why do you need Fulton's construction? Your equality only involves tangent bundles of smooth varieties. $\tilde{X} $ is a hypersurface in $B_p(\Bbb{P^n})$, where $p$ is the singular point of $X$. The formula for $[T_{\tilde{X} }]$ follows immediately from the normal bundle exact sequence for $\tilde{X}\subset B_p(\Bbb{P^n})$. $\endgroup$– abxCommented Mar 23, 2016 at 13:49
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