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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!

hyyyyy
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  • Hint on imitating Proposition V.3.4: Covering by trivializing affines of $Y$ and calculate the Cech cohomology to show $H^i(E,\mathcal{O}_E(n))=0$ for all positive $i,n$, where $E=\pi^{-1}Y=\mathbf{Proj}\mathcal{N}_Y$. – user10354138 Jun 06 '19 at 19:16
  • @user10354138 Can you explain how to calculate on that Cech complex to get the vanishing of the cohomology? I got the isomorphism $H^i(E,\mathcal{O}E(n)) \cong H^{i}(Y,\mathcal{I}^{n}{Y}/\mathcal{I}^{n+1}_{Y})$. – hyyyyy Jun 06 '19 at 19:49
  • For every affine $U=\operatorname{Spec}A\subseteq Y$, you get $\pi^{-1}U=\mathbb{P}A^{\operatorname{codim}Y}$, so $H^i(\pi^{-1}U,\mathcal{O}_E(n)\vert{\pi^{-1}U})=0$ for all $i,n>0$. Now Cech cohomology of the affines (intersection of affines is affine, etc.) gives the result. – user10354138 Jun 06 '19 at 19:55
  • @user10354138 I understand we can calculate from the open covering $\pi^{-1}U$, what I'm confused is that why the Cech complex gives vanishing cohomology. Is there any observation to see it's exact on $i>0$ terms? – hyyyyy Jun 06 '19 at 20:01

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