We know for $Y=\mathbb{P}^n$, the total space of the canonical sheaf $Tot(\omega_Y)$ is the resolution of $\mathbb{C}^{n+1}/\mathbb{Z}_n$ where the generator acts as scalar matrix of multiplying a primitive $n$-th root. I was told this result can be extended to certain types of del-pezzo surfaces but had a hard time to find the reference.

My question: When is the total space of the canonical bundle a resolution of singularity?

I would very much appreciate it if anyone can give some reference to this matter.

We know for $Y=\mathbb{P}^n$, the total space of the canonical sheaf $Tot(\omega_Y)$ is the resolution of $\mathbb{C}^{n+1}/\mathbb{Z}_{n+1}$ where the generator acts as scalar matrix of multiplying a primitive $n$-th root. I was told this result can be extended to certain types of del-pezzo surfaces but had a hard time to find the reference.

My question: When is the total space of the canonical bundle a resolution of singularity?

I would very much appreciate it if anyone can give some reference to this matter.