Let $S$ be a surface, $E$ be a curve on $S$, and $H$ be a hyperplane section of $S$. Let $a\in H^0(S,\mathcal O_S(H+(k-1)E))$, and $b\in H^0(S,\mathcal O_S(H+kE))$. Let $U$ be an open subset of $S$ on which $b$ does not vanish. I read that $\frac{a}{b}$ defines a section of $\mathcal O_U(-E)$, but I don't see why.
By definition of the invertible sheaf associtated to a divisor, I have $\mbox{div}(a)+H+(k-1)E\ge0$ and $\mbox{div}(b)+H+kE\ge0$, so my naive idea would be to substract the latter expression to the former, so that the left hand side is $\mbox{div}(\frac{a}{b})-E$, but I don't see how to show that this is effective.