Fake weighted projective spaces

Partly to support my qual preparation, partly to support my DRP student this semester, and partly for the sake of exposition, I am shortly going to begin writing a series of posts on toric geometry; with especial focus on toric Fano varieties and the classification problems thereof. As a sort of prelude, I offer a note I wrote about fake weighted projective spaces. Traditional weighted projective spaces have simplices as their toric polytopes, but it is not the case that all simplices give rise to weighted projective spaces. The additional toric varieties needed are fake weighted projective spaces: finite quotients of weighted projective spaces by a natural group action. This is a pleasing story in its own right, so I thought that I would tell it.

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