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Corollary 4.2.4 from Fulton's Intersection Theory gives a method for computing the Segre class of varieties, but in particular it allows computation of the Segre class for singular varieties.

Let $X$ be a proper subscheme of a variety $Y$, and let $\tilde{Y}$ be the blow-up of $Y$ along $X$. Then

$s(X,Y)=\sum_{k\geq 1}(-1)^{k-1}\eta_{\ast}(\tilde{X}^k)$

where $\eta :\tilde{X}\rightarrow X$ is the projection and $\tilde{X}$ is the exceptional divisor.

My question is this: are there any examples of explicit computations where this was used with a singular projective variety? I'm an undergrad, and as such I'm a little lost in the language of schemes. I have been unable to find an example where this was used in an explicit computation even though this seems to be the "go-to tool" for computing the Segre class of singular varieties.

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