On the Jordan-Hölder property for geometric derived categories

Abstract

We prove that the semiorthogonal decompositions of the derived category of the classical Godeaux surface X do not satisfy the Jordan-Hölder 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-Hölder property.

13 pages; Macaulay2 code available at http://www.math.uni-hamburg.de/home/boehning/research/JordanHoelderM2/M2scripts.html

References in corpus (5)

• Geometric Phantom Categories
• Determinantal Barlow surfaces and phantom categories
• On the derived category of the classical Godeaux surface
• Derived categories of Burniat surfaces and exceptional collections
• A simple counterexample to the Jordan-Hölder property for derived categories

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• Categorical dimension of birational automorphisms and filtrations of Cremona groups
• Discriminants and semi-orthogonal decompositions
• Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two
• Thick subcategories on weighted projective curves and nilpotent representations of quivers