The number of full exceptional collections for orbifold projective lines | Geometry and Dynamics Seminar

Abstract

The derived category of an orbifold projective line with positive Euler characteristic is equivalentto the one of an extended Dynkin quiver. For a Dynkin quiverObaidNaumanShammakhFakiehRingel gave a counting formula for the number of full exceptionalcollections in the derived category. The number coincides with the degree of the LyashkoLooijengamap for an ADE singularity. The equality of these numbers hints a consistency in some problems inBridgeland stability conditions and miror symmetry. in this talk, l will give a formula for the numberof full exceptional collections for an orbifold projective line, which can be regarded as ageneralization for Dynkin cases. Based on mirror symmetry, l will explain the relationship betweenthe number and the degree of the LyashkoLooienga map for the orbifold projective line. This talk isbased on a joint work with Yuuki Shiraishi and Atsushi Takahashi.