= Homogeneous rational sections of a twisting sheaf
{title2=$\Gamma(U,\mathcal O(m))=\{r:\deg r=m,\ r\text{ regular on }\pi^{-1}(U)\}$}
The local frame $X_i^m$ on $X_i\ne0$ identifies a section with a homogeneous rational function of degree $m$ on the punctured affine cone. Homogenizing a local rational expression and cancelling common factors writes every nonzero section as a quotient of coprime <homogeneous polynomials> $F/G$ with $\deg F-\deg G=m$. Conversely, such a rational function regular on the cone gives local coefficients $r/X_i^m$, which glue with the twisting sheaf's <transition functions of a vector bundle>.
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