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 ωX⊗Pα for all α∈Pic0X. We prove that this is equal to the relative base locus of ωX 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.