# Total space of canonical bundle on toric del pezzo surface

Let $X$ be a toric del pezzo surface with a full cyclic strong exceptional collection of line bundles, say $\mathbb{E}:=(E_1,\ldots,E_n)$, consider the total space of anti-canonical bundle $\omega_X^*$, denote by $Tot(\omega_X)$. There is a natural projection map: $Tot(\omega_X)\rightarrow X$ and for some special toric del pezzo surface, for example: https://arxiv.org/pdf/0904.0529.pdf. One is able to find that $Tot(\omega_X)$ is resolution of singularities of some quotient singularities of $\mathbb{C}^3$ by finite abelian group $G$. In this case, the tilting bundle on $Tot(\omega_X)$ is just $\bigoplus_i\pi^*(E_i)$. But my question is: Is there any example of toric del pezzo $X$ together with some cyclic strong exceptional collection of line bundles, such that $Tot(\omega_X)$ is resolution of singularities of some affine variety $Y$ quotient by some (abelian or non-abelian )group. I am sure I ever saw such examples on some papers, but I can not remember who wrote this paper. In my impression, his $Y$ is something like a quadric in $\mathbb{C}^4$. He obtained $Tot(\omega_X)$ as resolution of singularities of $Y$ quotient by $\mathbb{Z}_2$ or something similar, but I really dont remember the details. It looks like what he wrote was $X=\mathbb{P}^1\times\mathbb{P}^1$. But I really could not remember.

Thanks