The Stacks project

Exercise 111.44.7. Let $k$ be a field. Let $X = \mathbf{P}|^1 \times \mathbf{P}^1$ be the product of the projective line over $k$ with itself with projections $p : X \to \mathbf{P}^1_ k$ and $q : X \to \mathbf{P}^1_ k$. Let

Compute the dimensions of $H^ i(X, \mathcal{O}(a, b))$ for all $i, a, b$. Hint: Use Exercise 111.44.6.