Collecting various theories on toy examples: Projective space

Question

I am looking for text books/notes/papers/documents playing with toy examples: projective space, in particular, $P^{1}$. Because I think this is really a cute example. Although algebraic geometry on $P^{1}$ is comparatively simple, it gave inspirations to treat more general situtation. Precisely, I am looking for something including following topics:(but you can add whatever you want)

1. algebraic geometry of $P^{n}$, say, $Coh(P^{n})$, $D^{b}(Coh(P^{n}))$, say, exceptional collection,semi-orthogonal decomposition,stability conditions

2. Relation to representation theory, say, Hall algebra of $Coh(P^{n})$ and $D^{b}(Coh(P^{n}))$ and its relation to affine quantum group: $U_{q}(\hat{sl_{n+1}})$, representation theory of Kronecker quiver. Tilting theory. Weighted projective line and so on.

3. $D-module$ on flag variety of $U(sl_{2})$(or $P^{1}$) and so on..................

4. Add whatever you like.

5. Add whatever you like.

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Answer 1

Maybe I can answer this question by myself now. I did some literature research and find some papers and notes illustrating $P^{1}$ to establish various theory Lectures on Hall algebras The author talks about Hall algebra of coherent sheaves on $P^{1}$, relation with representation theory of $U_{q}(\hat{sl_{2}})$ and also The Hall algebra of the category of coherent sheaves on the projective line talks about similar facts.

Twisted rings of differential operators on the projective line and the Beilinson-Bernstein theorem. It is a master thesis by Koushik Panda. He established $P^{1}$(flag variety of $sl_{2}$) version of Beilinson-Bernstein localization. His treatment is very detailed.

t-stabilities and t-structures on triangulated categoriesillustrates classifications of t-structures on $D^{b}(Coh{P^{1}})$ Introduction to coherent sheaves on weighted projective lines by Chen-Xiaowu and Henning Krause. Very expository notes for coherent sheaves, Tilting theory, derived category of $P^{1}$

Answer 2

I'm far from an expert, but I think it's unlikely you're going to find much literature that focuses only on the projective line. But, for example, almost every algebraic geometry book I know uses the projective line and higher dimensional projective spaces as a basic example for everything. And, if a paper or book does not discuss explicitly these examples, then I think that is an opportunity for a student to work it out on his or her own.