No abstract

The nonexistence of semi-orthogonal decompositions in algebraic geometry is known to be governed by the base locus of the canonical bundle. We study another locus, namely the intersection of the base loci of line bundles that are isomorphic to the canonical bundle in the Néron-Severi group, and show that it also governs the nonexistence of semiorthogonal decompositions. As an application by using algebraically moving techniques, we prove that the bounded derived category of the i-th symmetric product of a smooth projective curve C has no nontrivial semi-orthogonal decompositions when the genus g(C) ≥ 2 and i ≤ g(C) − 1.

We prove that the bounded derived category of coherent sheaves of the Brill-Noether variety G r d (C) that parametrizing linear series of degree d and dimension r on a general smooth projective curve C is indecomposable when d ≤ g(C) − 1.[Tod21] Y. Toda. Derived categories of Quot schemes of locally free quotients via categorified Hall products.