A note on semiorthogonal indecomposability for some Cohen-Macaulay varieties

TL;DR

This paper extends a semiorthogonal indecomposability result from smooth Deligne-Mumford stacks to projective varieties with Cohen-Macaulay singularities, showing that certain singular curves have indecomposable derived categories.

Contribution

It generalizes a known theorem to Cohen-Macaulay singular varieties, demonstrating indecomposability of derived categories for a broader class of algebraic varieties.

Findings

All projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories.

Indecomposable categories of perfect complexes are established for these curves.

The extension applies the canonical bundle approach to singular varieties.

Abstract

In this short note, we observe that Theorem 3.1 in arXiv:1508.00682 for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes. Here weak indecomposablility refers to the admissibility of components.