Second Chern Class of Blow-up - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:24:09Z http://mathoverflow.net/feeds/question/93551 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93551/second-chern-class-of-blow-up Second Chern Class of Blow-up Marcello 2012-04-09T07:50:14Z 2012-04-09T13:36:14Z <p>Let X be a compact surface and $\tilde{X}$ its blowing-up, how can I show the formula $c_2(\tilde{X})=c_2(X)+1$?</p> http://mathoverflow.net/questions/93551/second-chern-class-of-blow-up/93560#93560 Answer by Harry for Second Chern Class of Blow-up Harry 2012-04-09T10:49:53Z 2012-04-09T10:49:53Z <p>Let me slightly expand on Dmitri's comment.</p> <p>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.)</p> <p>Then, if $X$ is irreducible and $n$-dimensional, by Gauss-Bonnet, we have $\deg c_n(X) = e_c(X)$.</p> <p>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 $e_c(Y) = e_c(X) +s$. </p> <p>This generalizes what you need.</p> http://mathoverflow.net/questions/93551/second-chern-class-of-blow-up/93565#93565 Answer by Liviu Nicolaescu for Second Chern Class of Blow-up Liviu Nicolaescu 2012-04-09T13:36:14Z 2012-04-09T13:36:14Z <p>Check <a href="http://mathoverflow.net/questions/92660%7D,%20" rel="nofollow">this question</a> where a more general situation is discussed.</p> <p><a href="http://mathoverflow.net/questions/92660" rel="nofollow">http://mathoverflow.net/questions/92660</a>, </p>