Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3 / a / Solution

Let ${\mathcal{K}}_X^{\ast }$ be the sheaf of nonzero meromorphic functions. A local equation for a divisor on a complex manifold is determined modulo a nowhere-zero holomorphic factor and hence defines a section of the divisor sheaf on a complex manifold ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$. Conversely, local representatives of such a section have quotients in ${\mathcal{O}}_X^{\ast }$, so their zero and pole orders agree on overlaps and define a divisor. The constructions are inverse: