Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the **deformation to the normal cone**:


**Verdier's version:** $\tilde{X}\_Y^\mathrm{Ver} := \operatorname{Bl}\_{Y\times \{0\}}(X\times \Bbb A^1\_k) - \operatorname{Bl}\_Y(X)$


**Fulton's version:** $\tilde{X}\_Y^\mathrm{Ful} := \operatorname{Bl}\_{Y\times \{0\}}(X\times \Bbb P^1\_k) - \operatorname{Bl}\_Y(X)$


Evidently, $\tilde{X}\_Y^\mathrm{Ver}$ is just $\tilde{X}\_Y^\mathrm{Ful}$ with the subvariety $X\times \{\infty\}\subseteq (X\times \mathbb P^1\_k)-(Y\times \{0\})\subseteq \tilde{X}\_Y^\mathrm{Ful}$ removed.



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> **My Question:** In what sort of situation is it necessary to use one version and not the other?
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Fulton's deformation to the normal cone vs Verdier's
*The following answer was emailed to me by Claude Sabbah. I have received his permission to post it here.*


Whenever you need to get some object on $Y$ from an object existing on the normal cone (in fact normal bundle in your case), you need to proceed by pushforward. As it is better to use a proper pushforward, the version $\tilde{X}\_Y^{\mathrm{Ful}}$ is best suited to the question. This is the way Fulton defines Chern classes for example. He needs to use a projection formula that only holds when the fibre of the projection is a projective space, not an affine space. 


On the other hand, Verdier is not interested in pushing forward to $Y$. The specialization of a perverse sheaf is defined on the normal bundle, and one recovers the nearby cycles by restricting to a section $Y \times \{1\}$, if it exists. Together with Fourier-Sato transform, this also gives the vanishing cycle functor. 


