Du Val Singularities and Dynkin diagrams references - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:22:54Z http://mathoverflow.net/feeds/question/106564 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106564/du-val-singularities-and-dynkin-diagrams-references Du Val Singularities and Dynkin diagrams references Qiao 2012-09-07T02:30:50Z 2012-09-07T22:23:18Z <p>May I ask whether there are good references for computing blowups of the Du Val Singularities? Also, how are these singularities related to the Dynkin diagrams?</p> http://mathoverflow.net/questions/106564/du-val-singularities-and-dynkin-diagrams-references/106577#106577 Answer by Sándor Kovács for Du Val Singularities and Dynkin diagrams references Sándor Kovács 2012-09-07T07:38:59Z 2012-09-07T07:38:59Z <p>See <a href="http://retro.seals.ch/digbib/view?rid=ensmat-001%3A1979%3A25%3A%3A300" rel="nofollow">Durfee, Alan H. (1979), "Fifteen characterizations of rational double points and simple critical points" , L'Enseignement Mathématique. Revue Internationale. IIe Série 25 (1): 131–163</a></p> http://mathoverflow.net/questions/106564/du-val-singularities-and-dynkin-diagrams-references/106630#106630 Answer by Jim Humphreys for Du Val Singularities and Dynkin diagrams references Jim Humphreys 2012-09-07T22:23:18Z 2012-09-07T22:23:18Z <p>For the geometric aspects in the first question, Alan Durfee's survey linked by Sandor is a good starting point. For the second question, it's probably necessary to get into the viewpoint of algebraic groups (in characteristic 0). Grothendieck foretold where the simple singularities should occur "in nature" in the algebraic group framework. This was proved by Brieskorn, whose student Peter Slodowy then filled in the picture. Slodowy's monograph <em>Simple Singularities and Simple Algebraic Groups</em> (Springer Lect. Notes. in Math. 815, 1980) is still a basic source, along with his various lecture notes and later papers. Here one sees the Dynkin diagrams intrinsically realized. At the same time the whole subject is treated in suitable generality in terms of the group structure.</p> <p>Maybe it's possible to understand your first question without going this far into the technical aspects of algebraic groups, but it's good to be aware of that direction along with traditional approaches to singularities.</p>