Cohomology of line bundles on the blowup of $\mathbb P^2$ Let $X$ denote the blowup of $\mathbb P^2$, $E$ the exceptional divisor, and $H$ the pullback of the hyperplane class.  How can I compute $H^0(X,mE+nH)$, $H^1(X,mE+nH)$, and $H^2(X,mE+nH)$ for $m,n \in \mathbb Z$?  If I'm working analytically, how can I think of these geometrically (e.g. "an element of $H^1(X,2E)$ is equivalent to a $1$-form on $\mathbb P^2$ such that...")?
 A: I assume you're only blowing up at one point, so here's at least a nice geometric description for $H^0$.
If $p$ is the point blown up to $E$, then $dH-E$ is the system of plane curves of degree $d$ passing through $p$, $dH-2E$ are those that have a double point at $p$, etc.  This works with any number of blownup points, and a good exercise is using this interpretation to find all 27 lines on a cubic surface (which is the blowup at 6 points)
A: Let $X$ be a smooth surface and $\pi: \widetilde{X}\rightarrow X$ be the blow up of $X$ at one point. By Hartshorne, Algebraic Geometry, Proposition V.3.4,
$$R^{i}\pi_{*}\mathcal{O}_{\widetilde{X}}=0$$ for all $i>0$ and $\pi_*\mathcal{O}_\widetilde{X}\cong\mathcal{O}_X$.
By the projection formula, for any divisor $D\subset X$, we have
$$H^{i}(\widetilde{X},\pi^{*}D)=H^{i}(X,D).$$
Now we come back to your question. If $X$ is the blow up of $\mathbb{P}^{2}$, $H^{i}(X,nH)=H^{i}(\mathbb{P}^{2},\mathcal{O}(n))$.
$$0\rightarrow \mathcal{O}_{X}(nH+(m-1)E)\rightarrow \mathcal{O}_{X}(nH+mE)\rightarrow \mathcal{O}_{E}(nH+mE)\rightarrow 0$$
Since $E(nH+mE)=mE^2=-m$, $\mathcal{O}_{X}(nH+mH)\cong \mathcal{O}_{\mathbb{P}^2}(-m)$. Then by taking the long exact sequence, we may relate the cohomology of $nH+(m-1)E$ and $nH+mE$.
Since we already know the cohomology of $nH$, we can inductively get the cohomology of $nH+mE$ for any $m,n\in \mathbb{Z}$.
