TITLE: (Second) Chern class of projective space, blown up in a linear subvariety
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: Since you are blowing up $\mathbb{P}^n$ along a linear subvariety, which in turn is a smoooth complete intersection in $\mathbb{P}^n$, you can compute its Chern classes very easily via a formula of Aluffi (Lemma 1.3 in http://www.math.fsu.edu/~aluffi/archive/paper348.pdf). After denoting the first Chern class of a hyperplane in $\mathbb{P}^n$ by $H$, $f:\tilde{\mathbb{P}^n}\to \mathbb{P}^n$ the blowup of $\mathbb{P}^n$ along a linear subvariety say of codimension $k$ and the exceptional divisor by $E$, then
$$
c(\tilde{\mathbb{P}^n})=\frac{(1+E)(1+f^*H-E)^k}{(1+f^*H)^k}f^*c(\mathbb{P}^n).
$$ 
(To get the individual Chern classes, just replace $H$ by $t\cdot H$ and $E$ by $t\cdot E$ in the expression above, expand the expression as a powerseries in $t$, then the coefficient of $t^m$ will be $c_m$.) Then if you want to compute any Chern number, take the product of the corresponding Chern classes and push them forward to $\mathbb{P}^n$, which amounts to just knowing the pushforwards of self-intersections of the exceptional divisor via the projection formula, which can be easily computed via Lemma 2.1 in http://arxiv.org/pdf/1211.6077v1.pdf.