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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.

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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 $\mathbb{F}_2$.

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  • $\begingroup$ Thank you very much. What are the preimages of $\mathcal{O}(e)$ and $\mathcal{O}_e$ $\endgroup$
    – Z.N
    Commented Sep 12, 2023 at 10:43
  • $\begingroup$ Note that any four consecutive objects in the list form a full exceptional collection. You can use it to decompose the objects of your interest and then translate to the other side using the action on the exceptional collection. $\endgroup$
    – Sasha
    Commented Sep 12, 2023 at 16:58

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