Let $S$ be a rational surface over $\mathbb{C}$. We denote by $\Delta$ the diagonal of $S \times S$ ($\Delta \simeq S$). Then we can consider the sheaf $\mathcal{O}_{S \times S}/I_{\Delta}^n$.
What are the rank and degree of $\mathcal{O}_{S \times S}/I^n_{\Delta}$?
If $S$ is smooth, there is an isomorphism $$ I^k/I^{k+1} \cong \Delta_*(I/I^2) \cong \Delta_*(S^k\Omega_S), $$ where $\Delta$ is the diagonal embedding $S \to S \times S$. Furthermore, the sheaf $\mathcal{O}_{S\times S}/I^n$ has a filtration by $I^k/I^n$ with associated graded factors $$ I^k/I^{k+1},\qquad 0 \le k \le n-1. $$ These two observations allow to compute the invariants of $\mathcal{O}_{S\times S}/I^n$.