What is the origin of and motivation for the notion fake weighted projective spaces? Could you please compare this notion to that of genuine weighted projective spaces, giving significant examples? Is there any good reference on fake lens spaces? (Thanks to the "user 15817").

What is the origin of and motivation for the notion fake weighted projective spaces? Could you please compare this notion to that of genuine weighted projective spaces, giving significant examples? Is there any good reference on fake weighted projective spaces as the one on fake lens spaces? (Thanks to the "user 15817").

How are they originated ? Could you please compare to the genuine weighted projective spaces  , giving significant examples  ? Is there any good reference as for fake lens spaces  ? (Thanks to the "user 15817").

What is the origin of and motivation for the notion fake weighted projective spaces? Could you please compare this notion to that of genuine weighted projective spaces, giving significant examples? Is there any good reference on fake lens spaces? (Thanks to the "user 15817").

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 !).

How are they originated ? Could you please compare to the genuine weighted projective spaces , giving significant examples ? Is there any good reference as for fake lens spaces ? (Thanks to the "user 15817").