Abstract

A fundamental question in algebraic geometry is when the minimals are really minimal. The first minimal means the canonical bundle of the variety is nef. The second minimal means the associated bounded derived category of coherent sheaves of the variety is semi-orthogonal indecomposible. In this talk, I will provide a criterion for semi-orthogonal indecomposability by the base point of para-canonical systems. As an application of the criterion, we prove that the bounded derived category of i-th symmetric power of a curve C is indecomposible if i less or equal g(C)-1. If time permits, I will mention the applications of para-canonical systems to the problems of Bridgeland stability conditions and nonexistence of phantom categories.