I'm having some trouble in computing the Čech cohomology for simple schemes (a smooth projective plane curve, for instance). In general, the procedure, apparently, is to find a good cover by affine open sets and then make some computations. However these computations deals with localization of the sections to smaller open sets (the intersection).
For instance, in exercise III 4.7 of Hartshorne, I need to get the degree genus formula for a smooth projective plane curves (not necessarilly irreducible), namely $C = V(f(x_0, x_1,x_2))$ of degree $d$, such that $C \cap [1 : 0: 0] = \emptyset$. Therefore, one can cover the curve with two affine open sets $U = C \cap \{x_1 \neq 0 \}$ and $V = C \cap \{ x_2 \neq 0 \}$ and, then compute $$\Gamma(U, \mathcal{O}_X) \oplus \Gamma(V, \mathcal{O}_X) \rightarrow \Gamma(U \cap V, \mathcal{O}_X)$$ But for $(s_U, s_V) = (g(x_0/ x_1, x_2/ x_1) + (f), h(x_0 / x_2, x_1 /x_2) + (f))$ in the left term, $\delta (s_U, s_V) = (g(x_0/ x_1, x_2/ x_1) + (f))|_{U \cap V} - (h(x_0 / x_2, x_1 /x_2) + (f))|_{U \cap V}= $ something that I don't know how to compute. Furthermore I could have used other local coordinates, namely $(g(x_0, x_2) + (f), h(x_0, x_1) + (f))$ and, then I would get $\delta (s_U, s_V) = g(x_0, 1) - h(x_0, 1) + (f(x_0, 1, 1))$ which seems wrong. So, how shoud I proceed?
Moreover, I would be glad if someone could show the same kind of computation in other schemes than the projective space.
Thanks in advance.