I'm stucking in the following exercise 3.1 from chapter V of Hartshorne's Algebraic geometry:
Let $X$ be a nonsingular projective variety, Y a nonsingular subvariety, and let $\pi : \tilde{X} \rightarrow X$ be the blowing up along Y. Show that $p_a(\tilde{X})\ =\ p_a(X)$.
I've tried the following approaches:
(1):imitate the proof of (3.4) where Hartshorne prove the result for blowing up a point on a smooth surface, in this case I need to calculate the cohomology of the conormal bundle:$H^{i}(Y,\mathcal{I}_{Y}/\mathcal{I}^{2}_{Y})$. I don't know how to continue from here.
(2):my another attempt is to use induction to a section $H$, by the short exact sequence $$ 0 \rightarrow \mathcal{O}_X(-H) \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_H \rightarrow 0 $$ $$ 0 \rightarrow \mathcal{O}_{\tilde{X}}(-\tilde{H}) \rightarrow \mathcal{O}_\tilde{X} \rightarrow \mathcal{O}_\tilde{H} \rightarrow 0 $$ use induction on $\tilde{H} \rightarrow H$, I want to show the euler characteristic of terms on left are the same, where I stucked.
I appreciate any ideas about the proof or any references for the proof, thank you!