2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61208We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the rational connectedness conjecture in [KoMiMo] which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group (now a Theorem of S. Takayama).Journal of Pure and Applied Algebra, to appear; 24 pagesAlgebraic Geometry14J28; 14F35text