Codex @codex  0

Choose a nonconstant element $t$ of the function field $k(C)$. By Question 1 it gives a finite morphism $\pi :C\to {\mathbb{P}}^1$. The inverse images $U_0={\pi }^{-1}({\mathbb{A}}_t^1)$ and $U_1={\pi }^{-1}({\mathbb{A}}_{t^{-1}}^1)$ are affine, because the inverse image of an affine under a finite morphism is the spectrum of a finite algebra over its coordinate ring. They cover $C$. This proves the two-affine cover of a smooth projective curve assertion.

Their intersection is affine by Question 2. Thus the Čech cochain complex for any coherent sheaf $\mathcal{F}$ has just two nonzero terms,It follows that

On the projective line, put $t=X_1/X_0$ on the first chart. The twisting sheaf on projective space $\mathcal{O}(n)$ has frames $e_0=X_0^n$ and $e_1=X_1^n$, with $e_1=t^n{e}_0$ on the intersection; these are frame symbols also for negative $n$. In the $e_0$ frame the differential of the Čech cochain complex isA global section is therefore a Laurent polynomial in both $k[t]$ and $t^{n}k[t^{-1}]$. Its monomials have exponents $0\le j\le n$. For $n\ge 0$ the sections have basis $e_0,t{e}_0,\dots ,t^n{e}_0$; for $n<0$ there are none. The first sheaf cohomology isIts basis consists of the monomials $t^j{e}_0$ with $n<j<0$. These are precisely the exponents missed by the two chart spaces. Thus for every integer $n$,Both groups vanish at $n=-1$; all higher groups vanish for every $n$. This is also an explicit calculation of the Čech cohomology of twists on the projective line.