On Nonexistence of Semi-Orthogonal Decompositions in Algebraic Geometry

We study the intersection of the base loci of line bundles that are isomorphic to the canonical bundle in the Néron–Severi group, and show that it also governs the nonexistence of semi-orthogonal decompositions. As an application, we show the bounded derived category of the i-th symmetric product of a smooth projective curve C has no nontrivial semi-orthogonal decompositions when the genus g(C)≥ 2 and i≤ g(C)-1. We show the indecomposability of derived categories of some examples of elliptic surfaces with pg(X)=0, and some examples of minimal surfaces of general type. An inequality involving phases of skyscraper sheaves for any Bridgeland stability condition is obtained.