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

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

Theorems & Definitions (26)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3: Kawatani-Okawa
• Theorem 1.4: Lin
• Theorem 1.5
• Corollary 1.6
• Remark 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10