Indecomposability of derived categories in families

TL;DR

This paper introduces a new approach to understanding when derived categories are indecomposable in families of algebraic varieties, using moduli spaces of semiorthogonal decompositions, with implications for various geometric examples.

Contribution

It proposes a novel method based on moduli spaces to study indecomposability of derived categories in families, connecting it to the behavior of the canonical base locus.

Findings

Established general results under a conjecture on the moduli space structure.

Analyzed explicit examples including symmetric powers of curves and Hilbert schemes.

Identified open cases and potential for future proofs using base locus analysis.

Abstract

Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a natural conjecture on the structure of the moduli space, we give both general results, and discuss interesting explicit examples of the behaviour of indecomposability in families, by relating it to the behaviour of the canonical base locus in families. These examples are symmetric powers of curves, certain regular surfaces of general type with large canonical base locus, and Hilbert schemes of points on surfaces. Indecomposability for symmetric powers of curves has been settled via other means, the other cases remain open and we expect that our analysis of the base locus will prove instrumental in finding unconditional proofs.