I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to this question, but I cannot really make much sense of the answer given there.

Write $\mathbb{P}^n:=\mathbb{P}^n_\Bbbk$ for projective space over some (algebraically closed) field $\Bbbk$ and assume that $X\subseteq\mathbb{P}^n$ is a linear subvariety, $\mathbb{P}^m\cong X$, say. I now consider the blow-up of $Y:=\mathbb{P}^n$ in $X$, yielding a blow-up diagram $$\begin{matrix} \tilde{X} & \xrightarrow{\;j\;} & \tilde{Y} \\ \hphantom{\scriptstyle g}\downarrow {\scriptstyle g} && \hphantom{\scriptstyle f}\downarrow {\scriptstyle f} \\ X &\xrightarrow{\;i\;} & Y \end{matrix}$$ My question is, what is the second chern class $c_2(\tilde Y):=c_2(\mathcal{T}_{\tilde{Y}})$ of the tangent sheaf of $\tilde{Y}$?

**Remark**: I am ultimately interested in the degree of $c_2(\tilde Y)c_1^{n-2}(\tilde Y)$. If I could understand the total chern class $c(\tilde Y)$, that would be even better.

**My thoughts so far:** The chern classes of $Y$ (and $X$) have well-known representation, and there is a formula for computing the chern classes of blown-up varieties in Fulton's book *Intersection Theory*, namely *Theorem 15.4*. For brevity, I will quote his *Example 15.4.3*, which gives a formula for $c_2$:

$$c_2(\tilde Y) = f^\ast(c_2(Y)) - j_\ast\left( (d-1) g^\ast(c_1(X)) + \tfrac{d(d-3)}{2} \zeta + (d-2) g^\ast(c_1(\mathcal{N})) \right)$$

Here, $\mathcal{N}=\mathcal{N}_{X/Y}$ is the normal bundle of $X$ in $Y$ and $\zeta$ denotes $c_1(\mathcal{O}_{\tilde{X}}(1))$.

From Fulton's *Example 3.2.12*, we know that $c_1(X)=(m+1-d)\cdot\xi$ and $c_1(\mathcal{N})=d\cdot\xi$ with $\xi = c_1(\mathcal{O}_X(1))$. Now, I am kinda stuck. I am not sure what the push-forwards and pullbacks really do - in particular, what is $g^\ast(\xi)$ in terms of $\zeta$? What is $f^\ast$, applied to the class of a hyperplane? What does $j_\ast$ do? More importantly, are these the right questions to ask?

Ultimately, I thought (hoped) it would be possible to express $c_2(\tilde Y)$ or even $c(\tilde Y)$ as a sum of intersections of "obvious" cycles in $\tilde Y$, possibly involving only the class of the (strict) transform of a hyperplane and the exceptional divisor.