Homogeneous rational sections of a twisting sheaf ($\Gamma (U,\mathcal{O}(m))=\{r:\mathrm{deg}r=m,r\text{ regular on }{\pi }^{-1}(U)\}$)

The local frame $X_i^m$ on $X_i\ne 0$ 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 $\mathrm{deg}F-\mathrm{deg}G=m$. 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.