Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of $X$ (i.e. of the tangent bundle of $X$) and the degrees of the Chern classes of $\widetilde{X}$?
Thanks.
Best Answer
For the first Chern class you get the simple formula
$$c_1(\tilde X)=p^*c_1(X)- (n-1)E$$ where $p:\tilde X \to X$ is the projection and $E$ the exceptional divisor.
In general the formula is more complicated and I'll refer you to Fulton's Intersection Theory, where the formula you require is given in Theorem 15.4.
In particular cases the relation may be quite simple: for example if $X$ is of dimension 3, it is just $c_2(\tilde X)=p^*(c_2(X)$ for the second Chern class, as proved in Griffiths-Harris's Principles of Algebraic Geometry, page 609.