Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories

Motivated by an indecomposability criterion of Xun Lin for the bounded derived category of coherent sheaves on a smooth projective variety X, we study the paracanonical base locus of X, that is the intersection of the base loci of \(\omega _X \otimes P_{\alpha }\) ${\omega }_X\otimes P_{\alpha }$ for all \(\alpha \in \textrm{Pic}^0 X\) $\alpha \in {\text{Pic}}^{0}X$ . We prove that this is equal to the relative base locus of \(\omega _X\) ${\omega }_X$ with respect to the Albanese morphism of X. As an application, we get that bounded derived categories of Hilbert schemes of points on certain surfaces do not admit non-trivial semi-orthogonal decompositions. We also have a consequence on the indecomposability of bounded derived categories in families. Finally, our viewpoint allows to unify and extend some results recently appearing in the literature.