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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...")?

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    All is not lost, however. Holomorphic differentials do capture cohomological information about a variety, the so-called "Algebraic de Rham cohomology" defined vaguely analogously to the way it is in diff. geom. But to compute it you actually need to use a spectral sequence with $E_2$ page $H^p(X,\Omega^q_X)$. On a smooth surface $X$, $\Omega^1_X$ is a vector bundle of rank 2, so without doing some more work, you can't immediately get at is cohomology using only the cohomology of divisors.2010-08-25

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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)