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

TL;DR

This paper investigates the paracanonical base locus and its relation to the Albanese morphism, demonstrating the indecomposability of derived categories for certain varieties and extending existing results in the field.

Contribution

It establishes the equality between the paracanonical base locus and the relative base locus of the canonical bundle, and applies this to prove indecomposability of derived categories for specific cases.

Findings

Paracanonical base locus equals the relative base locus of omega_X with respect to the Albanese morphism.

Bounded derived categories of certain Hilbert schemes lack non-trivial semi-orthogonal decompositions.

Indecomposability of derived categories extends to families of varieties.

Abstract

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 $\omega_X \otimes P_{\alpha}$, for all $\alpha \in \mathrm{Pic}^0 X$. We prove that this is equal to the relative base locus of $\omega_X$ 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.