Let X be a compact surface and $\tilde{X}$ its blowingup, how can I show the formula $c_2(\tilde{X})=c_2(X)+1$?

Let me slightly expand on Dmitri's comment. Let $X$ be a finite type separated $\mathbf{C}$scheme. Let $e_c(X)$ be the compactly supported Euler characteristic. (We consider the singular cohomology of $X$ with $\mathbf{Q}$coefficients.) Then, if $X$ is irreducible and $n$dimensional, by GaussBonnet, we have $\deg c_n(X) = e_c(X)$. Let $p:Y\to X$ be a proper birational surjective morphism. Let $s$ be the number of exceptional components of $p$. It is easy to see that $e_c(Y) = e_c(X) +s$. This generalizes what you need. 


Check this question where a more general situation is discussed. 

