I have a surface inside a toric variety $X$ and I would like to compute the first cohomology of its structure sheaf via the Cech complex, since I already know which cones of $X$ it hits (five two-dim., one three-dim.). I already computed the rings of the relevant cones and the respective local equations of the surface.
My problem is, that these rings have a number of variables depending on the number of elements in the Hilbert basis, so it already seems quite strange to add polynomial rings with different number of variables and relations (they're pretty big as well).
Is there any efficient way to construct the cech complex and compute the first cohomology group? Especially with Macaulay2?
Thanks for your help!
Regards
Richard