I am looking for some examples of computing the dimension of the first sheaf cohomology for smooth projective surfaces. To be more precisely, let $X$ be a smooth, projective surface. Let $D$ be an invertible coherent sheaf. I hope to know examples, beside of different kinds of vanishing theorem, in which people could really compute the dimension of the first cohomology $H^1(X, \mathcal{O}_X(D))$. Here by the word "compute", it could either mean to find a concrete integer, or it could mean to relate this dimension with some invariants of $X$ (or $D$ as well). However, I am not looking for an expression such as the Riemann-Roch formula, unless the $0$th and the $2$ed sheaf cohomology in the example can be computed concretely. Any comments, suggestions, or refereces are welcome.

I am looking for some examples of computing the dimension of the first sheaf cohomology for smooth projective surfaces. To be more precisely, let $X$ be a smooth, projective surface. Let $D$ be an invertible coherent sheaf. I hope to know examples, beside of different kinds of vanishing theorem, in which people could really compute the dimension of the first cohomology $H^1(X, \mathcal{O}_X(D))$. Here by the word "compute", it could either mean to find a concrete integer, or it could mean to relate this dimension with some invariants of $X$ (or $D$ as well). However, I am not looking for an expression such as the Riemann–Roch formula, unless the $0$th and the $2$nd sheaf cohomology in the example can be computed concretely. Any comments, suggestions, or references are welcome.