Let \(X\) be a nonsingular projective variety of any dimension, let \(Y\) be a nonsingular subvariety, and let \(\pi: \tilde{X} \rightarrow X\) be obtained by blowing up \(Y\). Show that \(p_a(\tilde{X})=p_a(X)\).
Solution
Put \[ n=\dim X. \] Since a blowup is birational, \[ \dim\widetilde X=n. \]
1. For every \(i\ge0\),\[ \boxed{ H^i(\widetilde X,\OO_{\widetilde X}) \cong H^i(X,\OO_X).} \]
This is Hartshorne V.3.4, applied to the blowup \[ \pi:\widetilde X=\operatorname{Bl}_Y X\longrightarrow X \] of the nonsingular projective variety \(X\) along the nonsingular centre \(Y\). It is exactly the blowup-cohomology statement cited by the source solution to V.3.1. In the surface case the same fact is recorded in FE-SRFBLOW as the invariance of \(\chi(\OO)\), \(p_g\), and \(q\).
2. The structure-sheaf Euler characteristic is unchanged:\[ \boxed{ \chi(\widetilde X,\OO_{\widetilde X}) = \chi(X,\OO_X).} \]
By definition, \[ \chi(Z,\OO_Z) = \sum_{i\ge0}(-1)^i \dim_k H^i(Z,\OO_Z) \] for a projective variety \(Z\). Step 1 identifies every summand for \(Z=\widetilde X\) with the corresponding summand for \(Z=X\), so the two alternating sums agree.
3. The arithmetic genera agree:\[ \boxed{p_a(\widetilde X)=p_a(X).} \]
For a nonempty projective scheme of dimension \(n\), the arithmetic genus is D-COHEULER \[ p_a(Z)=(-1)^n\bigl(\chi(Z,\OO_Z)-1\bigr). \] The varieties \(X\) and \(\widetilde X\) have the same dimension \(n\), and step 2 gives equality of their structure-sheaf Euler characteristics. Substitution into the displayed definition gives \[ p_a(\widetilde X) = (-1)^n\bigl(\chi(\widetilde X,\OO_{\widetilde X})-1\bigr) = (-1)^n\bigl(\chi(X,\OO_X)-1\bigr) = p_a(X). \]
4. Q.E.D.
Steps <1>1–<1>3 prove the asserted invariance of the arithmetic genus under blowing up a nonsingular centre.