Let $k$ be an algebraically closed field of characteristic zero. Let $\Sigma$ be the fan in $\mathbb{R}^2$ consisting of three cones, cone generated by $e\_1,e\_2$,cone generated by $e\_2,-e\_1-2e\_2$ and the cone generated by $-e\_1-2e\_2,e\_1$.Let $U$ be the toric variety associated to the cone in $\mathbb{R}^2$ which is generated by $e\_1,e\_2$. Then it is easy to show that $U$ is $\mathbb{A}^2\_k$. I am trying to compute all the $G$-theory groups of the toric variety associated to the fan $\Sigma$, which is the weighted projective space $\mathbb{P}(1,1,2)$. Here,$G$-theory means the algebraic $K$-theory of the abelian category of coherent sheaves.I am thinking of using the $G$-theory localization sequence induced by the homotopy fibration $G(Z)\rightarrow G(X)\rightarrow G(U)$, where $X$ is $\mathbb{P}(1,1,2)$, $Z$ is the complement of $U$ in $X$.
Since $G\_n(U)\cong G\_n(k)$ for all $n$, after passing to the reduced $G$-theory localization sequence, we get that $\tilde{G}\_{n}(Z)\rightarrow \tilde{G}\_n(X)$ are isomorphisms for all $n$. Here,I defined $\tilde{G}\_n(U)$ to be the cokernel of the homomorphism $G\_n(k)\rightarrow G\_n(U)$ induced by the structure morphism of the $k$-variety $U$. $\tilde{G}\_n(X)$ is defined similarly.
So now I need to understand the closed subvariety $Z$ of $X$, in order to compute the $G$-theory groups of $X$.
Also, are there known methods of calculating the $G$-theory groups of weighted projective spaces?



How to compute the $G$-theory of the weighted projective space $\mathbb{P}(1,1,2)$?
If $X$ is a projective rational normal surface then
$$
G\_0(X) \cong \mathbb{Z} \oplus \mathrm{Cl}(X) \oplus \mathbb{Z},
$$
see, e.g., Lemma 4.2 in [Karmazyn, Joseph; Kuznetsov, Alexander; Shinder, Evgeny. Derived categories of singular surfaces. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 2, 461--526]. Since
$$
\mathrm{Cl}(\mathbb{P}(1,1,2)) \cong \mathbb{Z},
$$
it follows that $G\_0(\mathbb{P}(1,1,2)) \cong \mathbb{Z}^3$.


