Cellular resolutions of the diagonal and exceptional collections for toric D-M stacks (Reginald Anderson, CMC)

Beilinson gave a resolution of the diagonal for complex projective space, which gives a strong, full exceptional collection of line bundles as a generating set for the derived category of coherent sheaves. Bayer-Popescu-Sturmfels generalized Beilinson’s resolution of the diagonal by giving a cellular resolution of the diagonal for a proper subclass of smooth toric varieties which they called “unimodular.” In joint work with Gabe Kerr, we extended this resolution of the diagonal to smooth projective toric varieties by showing that the cokernel of Bayer-Popescu-Sturmfels’ resolution is torsion with respect to the irrelevant ideal. In this talk, we show that Bayer-Popescu-Sturmfels’ resolution yields a strong, full, exceptional collection of line bundles for unimodular projective toric surfaces, and that our extended resolution of the diagonal yields a strong, full exceptional collection of line bundles for a smooth, non-unimodular projective toric surface.