TITLE: Is the algebraic Grothendieck group of a weighted projective space finitely generated ?
QUESTION [6 upvotes]: This is to be confronted with Joseph Gubeladze' paper : "Toric varieties with huge Grothendieck group" !

REPLY [3 votes]: In our paper with N. Pavic we have proved that over an algebraically closed field of char. 0, if the weights $a_0, \dots, a_n$ are coprime, so that singularities of  $\mathbb{P}(a_0, \dots, a_n)$ are isolated, then $K_0(\mathbb{P}(a_0, \dots, a_n)) = \mathbb{Z}^{n+1}$, see Application 3.2 in 
https://arxiv.org/pdf/1809.10919.pdf
Our proofs rely on comparison of $K_0(X) = K_0(Perf(X))$ with $G_0(X) = K_0(D^b(X))$ using $K$-groups of Orlov's singularity category and completion arguments. The same methods apply to compute $K_0(X)$ for any quasi-projective variety $X$ with isolated quotient singularities.