What is the meaning of "fake" for these spaces (both of them) ? Is it possible to have some explicit examples ? Miles Reid seems to be , involuntarily, behind the existence of fake weighted projective spaces . I red that, studying toric morphisms (contraction of extremal rays (Mori theory)), he identified, by mistake, some related fibres as WPS's . This would give birth to "fake" WPS's !. How much is this true ? Could you please illustrate ? (Thanks !).