Christian Böhning

On the derived category of the classical Godeaux surface

Abstract We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z / 5 Z -quotient of the Fermat quintic surface in P 3 . This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z 11 ⊕ Z / 5 Z . In particular, the result answers Kuznetsov’s Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E 8 . We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.

Determinantal Barlow surfaces and phantom categories

We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsovs results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category.

Isoclinism and stable cohomology of wreath products

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups (see Theorem 11). Moreover, we show that the stable cohomology of the n-fold wreath product \(G_{n} = \mathbb{Z}/p \wr \ldots \wr \mathbb{Z}/p\) of cyclic groups \(\mathbb{Z}/p\) is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.

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–Holder 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–Holder property.

Birationally isotrivial fiber spaces

We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will provide numerical evidence for potential restrictions on tuples of dynamical degrees realisable on general cubic fourfolds which can be violated on the projective four-space.We prove that a family of varieties is birationally isotrivial if all the fibers are birational to each other.

Stable cohomology of alternating groups

AbstractWe determine the stable cohomology groups (

The Rationality of the Moduli Space of Curves of Genus 3 after P. Katsylo

H_S^i \left( {{{\mathfrak{A}_n ,\mathbb{Z}} \mathord{\left/ {\vphantom {{\mathfrak{A}_n ,\mathbb{Z}} {p\mathbb{Z}}}} \right. \kern-\nulldelimiterspace} {p\mathbb{Z}}}} \right)

Birational Properties of Some Moduli Spaces Related to Tetragonal Curves of Genus 7

of the alternating groups