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This is to be confronted with Joseph Gubeladze' paper : "Toric varieties with huge Grothendieck group" !

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By "Algebraic Grothendieck group", do you mean the Grothendieck group of vector bundles? –  David Speyer Jun 11 '13 at 17:40
Yes , it is the Grothendieck group of vector bundles. The one of coherent sheaves is well-known. –  Al-Amrani Jun 11 '13 at 17:51
It is mentioned in the above paper that it was the initial idea to construct such examples with huge $K_0$ among weighted projective spaces, but so far without success. –  Dietrich Burde Jun 11 '13 at 18:53
Yes, that is right. But recently, Adam Massey ( KH-Theory of Complete Simplicial Toric Varieties and Algebraic K-theory of Weighted Projective Spaces ) obtained some progress in very particular case of weights, that is (1,...,1,q). –  Al-Amrani Jun 11 '13 at 19:25
In other words, the question is an open problem that looks well known among the people that know this sort of problems. Notice there is a minisection in the FAQ mathoverflow.net/faq about this. –  Mariano Suárez-Alvarez Jun 11 '13 at 19:40

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Adam Massey showed that K°(P(1,...,1,q)) = K°(P(1,...,1)).On the other hand P(1,...,1,q) is the cone with wertex (0, ...,0,1) which projects the Veronese variety Vq. Who knows any other particular nice geometrical exemples (small dimensions) of weighted projective spaces whith finitely generated algebraic GROTHENDIECK group (vector bundles) ?

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