TITLE: Why are coherent sheaves on $\Bbb P^1$ derived equivalent to representations of the Kronecker quiver?
QUESTION [17 upvotes]: I'm looking for an explanation or a reference to why there is this equivelence of triangulated categories: $${D}^b(\mathrm {Coh}(\Bbb P^1))\simeq {D}^b(\mathrm {Rep}(\bullet\rightrightarrows \bullet))$$
It is my understanding that the only reason why $\Bbb P^1$ appears at all is because it is used to index the regular irreducible representations of the Kronecker quiver. I have also heard that this equivalence can be used to understand the geometry of $\Bbb P^1$. 
I suppose more generally, adding arrows to the Kronecker quiver gives a similar result for $\Bbb P^n$.

REPLY [8 votes]: I'd just like to say thank you for the answers I have gotten for this question. After doing some more reading I have come across a few results which I shall summarise here.

Beilinson [1] gives the result that for projective space $\Bbb P^n$ the line bundles $\mathcal O_{\Bbb P^n}, \dots,\mathcal O_{\Bbb P^n}(n)$ freely generate $D^b(\mathrm{Coh}(\Bbb P^n))$.
Bondal [2] generalises this by observing that if line bundles $L_0,\dots,L_r$ freely generate $D^b(\mathrm {Coh} (X))$, then there is an equivilence of triangulated categories given by $${\operatorname{{\bf R}Hom}}_{\mathcal O_X}(\oplus_i L_i, -):D^b(\mathrm {Coh} (X))\longrightarrow D^b(\mathrm {mod}_A)$$ 
where $\mathrm {mod}_A$ is the category of finitely generated right modules over the algebra $A = \mathrm{End}(\oplus_i L_i)$.
King [3,4] shows that Hirzebruch surfaces $\Bbb F_n $ and the smooth Fano toric surfaces have a collection of line bundles that generate $D^b(\mathrm{ Coh}(X))$ as above. To do this, King showed that each of these toric surfaces is a fine moduli space of $\theta$-stable representations of a quiver with relations $Q$ whose path algebra is the endomorphism algebra $A$. The tautological bundles of the moduli space are the line bundles that generate $D^b(\mathrm{Coh}(X))$.
Craw-Smith [5] generalise King's construction to projective toric varieties of arbitrary dimension. They show that $X$ is isomorphic to a component of the fine moduli space of $\theta$-stable representations of a quiver with relations. (I think this quiver is the Bondal quiver?)
Craw-Winn [6] further generalise this result to Mori-dream spaces.

References:
[1] A. Beilinson. Coherent sheaves on $\Bbb P^n$ and problems in linear algebra. Funktsional. Anal. i Prilozhen., 12 (1978) 68-69.
[2] A. Bondal. Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 45 (1989) 25-44.
[3] A. King. Moduli of representations of finite-dimensional algebras. Quarterly J. Math. Oxford 45 (1994) 515-530.
[4] A. King. Tilting bundles on some rational surfaces. Unpublished article available from Alastair King's homepage http://people.bath.ac.uk/masadk/papers/ (near the bottom), (1997).
[5] A. Craw and G.G. Smith. Projective toric varieties as fine moduli spaces of quiver representations American Journal of Mathematics, 130 (2008) 1509-1534., arXiv:0608183
[6] A. Craw and D. Winn. Journal of Pure and Applied Algebra 217 (2013) 172-189 arXiv:1104.2490
Interestingly most of the authors names sound like mine.