Del Pezzo surfaces with log-terminal singularities. II

If $Z$ is a del Pezzo surface with log-terminal singularities of index dividing $k$ and $\sigma\colon Y\to Z$ the minimal resolution of singularities of $Z$, we prove the inequality $\operatorname{rk\,Pic}Y<Ak^{7/2}$, where $A$ is an absolute constant. It follows from this that for fixed $k$ there are only a finite number of possible intersection graphs of all exponential curves on $Y$. In Part I these results were obtained under a certain restriction on the singularities.
The proof uses methods taken from the theory of reflection groups in Lobachevsky space.
Bibliography: 14 titles.