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 CG(D) for each fixed maximal decomposition D. 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.

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

“…Such an analog would be the best candidate for a birational invariant. It turns out that such a component is not well defined in general; this is due to the failure of the Jordan–Hölder property for semiorthogonal decompositions, see . However, we can define the Griffiths–Kuznetsov component of a del Pezzo surface as follows.…”

A refined derived Torelli theorem for enriques surfaces, II: the non-generic case

“…where for the second one we use (2.3). The third isomorphism follows from the fact that S 2 X (F) ∼ = F [4] and S Ku(X ,L) (F) ∼ = F [3]. The fourth is a consequence of ζ !…”

“…where the first isomorphism is [20, Lemma 2.6], the second one is Remark 2.3 (i), the third one follows from the the fact that S 2 X ∼ = [4] and the fourth one follows from the observation that ζ ! K (S X (L i 1 )) = 0 (this by (1.1) and noticing that S X (L i 1 ) ∈ K ⊥ ).…”

“…Roughly, such a property predicts that if X is a smooth projective variety then the semiorthogonal decompositions of D b (X ) are essentially unique, up to reordering of the components and equivalence. It is worth recalling that we already know counterexamples to such a property [4,18].…”

Derived categories of Gushel–Mukai varieties

“…If the set of Griffiths components of did not depend on the choice of maximal semiorthogonal decomposition, then it would be a birational invariant [Kuz16b, Lemma 3.10]; in particular, it would be empty if is rational of dimension at least . Unfortunately, there are examples showing this is not true (see [Kuz16b, § 3.4], [BGS14]). It may be possible to salvage the situation by modifying the definition of a Griffiths component (some possibilities are discussed in [Kuz16b, § 3.4]), but this remains an important question.…”