Weighted projective lines and applications

Abstract

This survey covers the topics of my mini-course on “Weighted projective lines” given at the Workshop of ICRA XIV (XIV International Conference at Workshop, Tokyo, August 6–15, 2010). Weighted projective lines, and their defining equations, have a long history going back to Klein and Poincaré. Accordingly their study has a high contact surface with many mathematical subjects, classical and modern. Among the many related subjects we mention representation theory of algebras and groups, invariant theory, function theory, orbifolds, 3-manifolds, singularities and the study of nilpotent operators. Since the formal definition of the category of coherent sheaves by W. Geigle and the author in 1987, substantial progress has been made by a number of authors. As a recent application, the analysis of the singularity category of triangle singularities, Kleinian and Fuchsian singularities is covered in some detail. In the center of this analysis is the structure of the corresponding stable categories of vector bundles. The study is in the spirit of Buchweitz (1987) and Orlov (2005) and concerns work in progress with Kussin and Meltzer, recent work by Kajiura, Saito, and Takahashi (2007, 2009), and joint work with J. A. de la Peña (2006). These methods are further applied to the study by C. M. Ringel and M. Schmidmeier (2008) on the invariant subspace problem for nilpotent operators.