DELPEZZO SURFACES WITH LOG-TERMINAL SINGULARITIES .1.

A new method is applied to the study of del Pezzo surfacesZ with log-terminal singularities, taken from the theory of reflection groups in Lobachevsky space.This method yields bounds on the Picard number(Y) of a minimal resolution Y ofsingularities of Z, assuming that the indices or the multiplicities of the singularities ofZ are bounded, and under an extra (conjecturally inessential) condition of generality on the singularities of Z. © 1990 American Mathematical Society.