Tilting on singular rational projective curves

Abstract: My talk is based on a joint work in progress with Yuriy Drozd. Our main result is the following: let X be a reduced rational projective curve of any genus having only nodes or cusps as singularities. Then there exists a finite-dimensional algebra A  of homological dimension two and a fully faithful functor  from the triangulated category Perf(X) of perfect complexes on X in the derived category D^b(A-mod) of finite dimensional modules over A. In the case of  degenerations of elliptic curves, this leads to a particularly nice class of algebras called "gentle".  From one side, our approach  brings a new light on the representation theory of gentle algebras, on the other it  suggests a new approach to  some open problems about coherent sheaves on degenerations of elliptic curves. At the end,  I shall discuss a certain generalization of our technique on the case of K3 surfaces.