On the Jordan-H\"older property for geometric derived categories

TL;DR

This paper demonstrates that the derived category of the Godeaux surface and certain rational fourfolds have semiorthogonal decompositions that do not satisfy the Jordan-H"older property, revealing multiple maximal decompositions with different components.

Contribution

It proves the failure of the Jordan-H"older property for semiorthogonal decompositions of specific algebraic surfaces and fourfolds, introducing new examples and methods.

Findings

Godeaux surface's derived category has two maximal exceptional sequences of different lengths.

Different maximal decompositions of the same derived category can have non-isomorphic Clemens-Griffiths components.

Examples of rational fourfolds with non-unique semiorthogonal decompositions are provided.

Abstract

We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan-H\"older property. More precisely, there are two maximal exceptional sequences in this category, one of length 11, the other of length 9. Assuming the Noetherian property for semiorthogonal decompositions, one can define, following Kuznetsov, the Clemens-Griffiths component for each fixed maximal decomposition. We then show that D^b (X) has two different maximal decompositions for which the Clemens-Griffiths components differ. Moreover, we produce examples of rational fourfolds whose derived categories also violate the Jordan-H\"older property.