The vector bundle of an hypersurface

Question

Suppose that $X$ is a compact complex variety and $V \subset X$ an irreducible hypersurface.
Let $\{U_{\alpha}\}_{\alpha \in I}$ an open covering of X. With $s_{\alpha}$ i denote the local equation of $V$ in $U_{\alpha}$.
I know that $V$ is an hypersurface so $V$ is a Weil divisor on $X$. On the other hand, due to $X$ is compact, $V$ define also a Cartier divisor.
How can i express the Cartier divisor of $V$ or the cocycle of $V$ in the vector space $H^1(X,O^*)$ ?

Answer 1

You are looking for maps from $U_{\alpha\beta} = U_\alpha \cap U_\beta \to \mathcal{O}^*$ that naturally arise from the hypersurface. All you have given are the functions $s_\alpha : U_\alpha \to \mathbb{C}$ which vanish along $V$. What happens if you look at the quotients $s_\alpha/s_\beta$?