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Motivated by an indecomposability criterion of Xun Lin for the boundedderived category of coherent sheaves on a smooth projective variety $X$, westudy the paracanonical base locus of $X$, that is the intersection of the baseloci of $\omega_X \otimes P_{\alpha}$, for all $\alpha \in \mathrm{Pic}^0 X$.We prove that this is equal to the relative base locus of $\omega_X$ withrespect to the Albanese morphism of $X$. As an application, we get that boundedderived categories of Hilbert schemes of points on certain surfaces do notadmit non-trivial semi-orthogonal decompositions. We also have a consequence onthe indecomposability of bounded derived categories in families. Finally, ourviewpoint allows to unify and extend some results recently appearing in theliterature.

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