  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.
