I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following blow-up diagram:
$$\begin{matrix} E & \xrightarrow{\;j\;} & \tilde{Y} \\
\hphantom{\scriptstyle g}\downarrow {\scriptstyle g} &&
\hphantom{\scriptstyle f}\downarrow {\scriptstyle f} \\
X &\xrightarrow{\;i\;} & Y
 \end{matrix}$$
Denote by $P$ the proper transform, under $f$, of a hyperplane in $Y$.


I am trying to calculate $c\_2(\tilde Y)$, and with the help of [a post here at MO](https://mathoverflow.net/questions/90409/second-chern-class-of-projective-space-blown-up-in-a-linear-subvariety), I thought that I had figured it out. However, I wanted to do a quick check if nothing went wrong, but something *did* go wrong. Assume $n=4$ and $d=2$, then we can use the formula from Fulton's book in Example 15.4.3 to get
$$c\_2(\tilde Y)=f^\ast c\_2(Y) - j\_\ast g^\ast c\_1(X) - E^2.$$
By [the answer to my question](https://mathoverflow.net/questions/90409/second-chern-class-of-projective-space-blown-up-in-a-linear-subvariety) by Johannes Nordström, we can write $j\_\ast g^\ast c\_1(X)=3(E^2 + EP)$ and this yields
$$c\_2(\tilde Y)=10P^2 - 3EP - 4E^2.$$
Now, we also know from the same example in Fulton's book that $c\_1(\tilde Y)=f^\ast c\_1(Y) - E=5P-E$. Since the blow-up map $f$ is finite of degree one, the degrees of $c\_1^2(\tilde Y)c\_2(\tilde Y)$ and $c\_1^2(Y)c\_2(Y)$ should coincide. Because I was unsure of the calculation, I asked [a second question](https://mathoverflow.net/questions/90575/intersection-powers-of-the-exceptional-divisor-and-the-transform-of-a-hyperplane) and obtained the answer that
$$P^{n-b} E^b = (-1)^{b-1+n-d} \cdot \binom{b-1}{n-d}$$
Now, we can put this all together and obtain
$$\begin{align\*}
c\_1^2(\tilde Y)c\_2(\tilde Y) &= (5P-E)^2(10P^2 - 3EP - 4E^2)
\\&= 250P^4 - 175P^3E - 60P^2E + 37PE^3 - 4E^4
\\&= 250 + 37 + 12 = 299,
\end{align\*}$$
but $c\_1^2(Y)c\_2(Y)=(4+1)^2\cdot\frac{4(4+1)}{2} = 250$. 


I do not know where the mistake is, since I find both of the answers I received very convincing, but I cannot find a flaw in my calculation either, nor do I doubt Fulton.



Calculating chern numbers yields a contradiction, why?
Note that $P^2 = 0$, since we blow up the self-intersection of a hyperplane.


The pull-back of the hyperplane class is $P+E$, so $c\_1(\tilde Y) = 5P + 4E$, and $c\_2(\tilde Y) = 10(P+E)^2 - 3EP - 4E^2 = 17EP + 6E^2$.


This still does not yield $c\_1(\tilde Y)^2 c\_2(\tilde Y) = 250$ (I think it's $512 - 3\cdot 96 = 224$), but I don't see why this Chern number should be preserved by the blow-up.


