On nonexistence of semi-orthogonal decompositions in algebraic geometry

The nonexistence of semi-orthogonal decompositions in algebraic geometry is known to be governed by the base locus of the canonical bundle. We study another locus, namely the intersection of the base loci of line bundles that are isomorphic to the canonical bundle in the N\'{e}ron-Severi group, and show that it also governs the nonexistence of semi-orthogonal decompositions. As an application by using algebraically moving techniques, we prove that 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 $g(C)\geq 2$ and $i\leq g(C)-1$. We prove indecomposability of derived categories of some examples of elliptic surfaces with $P_{g}(X)=0$, and some natural examples of minimal surfaces of general type. Finally, an inequality involving phases of skyscraper sheaves for any Bridgeland stability condition is obtained.