Let $\tilde{X}$ be a desingularization of a normal surface $X$. The group Pic$\left(\tilde{X}\right)$ is provided with an order relation $L\underset{\_}{\gg }0$, defined by $L$. $V\le 0$ for any effective exceptional divisor $V$. Comparing to the usual order relation we define the ceiling of $L$ which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...