# Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface

Let $$k$$ be an algebraically closed field of $$\text{ch}(k) =0$$.

Let $$\mathbb{P}(1,1,2)$$ be the weighted projective plane of weight $$(1,1,2)$$ as a stack. Let $$\mathbf{P}(1,1,2)$$ be the weighted projective plane of weight $$(1,1,2)$$ as a scheme. Let $$\mathbb{F}_2$$ be the Hirzebruch surface of degree 2.

$$\mathbf{P}(1,1,2)$$ has a unique singularity and $$\mathbb{F}_2$$ is the minimal resolution.

From derived McKay correspondence, we have an equivalence of derived categories $$\Phi:D^b(\text{Coh}(\mathbb{P}(1,1,2))) \overset{\simeq}{\rightarrow} D^b(\text{Coh}(\mathbb{F}_2)).$$

Question

How is this functor $$\Phi$$ defined?

For example, what are the images of canonical line bundles $$\mathscr{O}_{\mathbb{P}(1,1,2)}(i)$$ by $$\Phi$$ ? Of course, I am also interested in the image of any other sheaf.

Any comment is welcome. Thank you.

One choice of $$\Phi$$ acts as $$\mathcal{O}_{\mathbb{P}(1,1,2)}(2i) \mapsto \mathcal{O}_{\mathbb{F}_2}(i(e + 2f)),$$ $$\mathcal{O}_{\mathbb{P}(1,1,2)}(2i+1) \mapsto \mathcal{O}_{\mathbb{F}_2}(i(e + 2f)+f),$$ where $$e$$ is the exceptional section and $$f$$ is the fiber of $$\mathbb{F}_2$$.
• Thank you very much. What are the preimages of $\mathcal{O}(e)$ and $\mathcal{O}_e$