Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and $\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an integer. Are there "formulas" for the sheaves $R^i\pi_*(\mathcal{O}_{\tilde{X}}(nE))$? Are there simple formulas?
1 Answer
Yes, sure, one can use inductively exact sequences $$ 0 \to \mathcal{O}_{\tilde{X}}((n1)E) \to \mathcal{O}_{\tilde{X}}(nE) \to \mathcal{O}_E(nE) \to 0 $$ to prove that $$ R\pi_*\mathcal{O}_{\tilde{X}}(nE) \cong I_Z^{n} $$ for all $n \le 0$. Similarly, for $n \ge 0$ one can check that $$ R^0\pi_*\mathcal{O}_{\tilde{X}}(nE) \cong \mathcal{O}_X $$ that $$ R^i\pi_*\mathcal{O}_{\tilde{X}}(nE) = 0 $$ for $1 \le i \le \mathrm{codim}(Z)  2$, and $$ R^{\mathrm{codim}(Z)  1}\pi_*\mathcal{O}_{\tilde{X}}(nE) $$ is an explicit sheaf supported on $Z$.

$\begingroup$ Thank you very much. Apparently there are no general results about this. $\endgroup$ Commented Apr 30, 2023 at 5:51