Collecting various theories on toy examples: Projective space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:43:30Z http://mathoverflow.net/feeds/question/17142 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17142/collecting-various-theories-on-toy-examples-projective-space Collecting various theories on toy examples: Projective space Shizhuo Zhang 2010-03-04T22:38:44Z 2010-03-29T13:15:06Z <p>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)</p> <ol> <li><p>algebraic geometry of $P^{n}$, say, $Coh(P^{n})$, $D^{b}(Coh(P^{n}))$, say, exceptional collection,semi-orthogonal decomposition,stability conditions</p></li> <li><p>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.</p></li> <li><p>$D-module$ on flag variety of $U(sl_{2})$(or $P^{1}$) and so on..................</p></li> <li><p>Add whatever you like.</p></li> <li><p>Add whatever you like.</p></li> </ol> <p>..................................................</p> http://mathoverflow.net/questions/17142/collecting-various-theories-on-toy-examples-projective-space/17184#17184 Answer by Deane Yang for Collecting various theories on toy examples: Projective space Deane Yang 2010-03-05T15:16:47Z 2010-03-05T15:16:47Z <p>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.</p> http://mathoverflow.net/questions/17142/collecting-various-theories-on-toy-examples-projective-space/19725#19725 Answer by Shizhuo Zhang for Collecting various theories on toy examples: Projective space Shizhuo Zhang 2010-03-29T13:15:06Z 2010-03-29T13:15:06Z <p>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 <a href="http://arxiv.org/abs/math/0611617" rel="nofollow">Lectures on Hall algebras</a> The author talks about Hall algebra of coherent sheaves on $P^{1}$, relation with representation theory of $U_{q}(\hat{sl_{2}})$ and also <a href="http://arxiv.org/abs/math/9906037v1" rel="nofollow">The Hall algebra of the category of coherent sheaves on the projective line</a> talks about similar facts.</p> <p><a href="http://web.maths.unsw.edu.au/~danielch/thesis/koush.pdf" rel="nofollow"> Twisted rings of differential operators on the projective line and the Beilinson-Bernstein theorem.</a> 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.</p> <p><a href="http://iopscience.iop.org/1064-5632/68/4/A04/pdf/IZV_68_4_A04.pdf?ejredirect=migration" rel="nofollow">t-stabilities and t-structures on triangulated categories</a>illustrates classifications of t-structures on $D^{b}(Coh{P^{1}})$ <a href="http://arxiv.org/abs/0911.4473" rel="nofollow">Introduction to coherent sheaves on weighted projective lines</a> by Chen-Xiaowu and Henning Krause. Very expository notes for coherent sheaves, Tilting theory, derived category of $P^{1}$</p>