Hans-Christian Graf von Bothmer

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.

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.

Algorithmic Construction of Hurwitz Maps

We describe an algorithm that, given a k-tuple of permutations representing the monodromy of a rational map, constructs an arbitrarily precise floating-point complex approximation of that map. We then explain how it has been used to study a problem in dynamical systems raised by Cui.

The transcendental lattice of the sextic Fermat surface

We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.

Focal Values of Plane Cubic Centers

We prove that the vanishing of 11 focal values is not sufficient to ensure that a complex plane cubic system has a complex center. This is done by finding a complex cubic system with a high order weak focus using an extensive computer search.We prove that the vanishing of 11 focal values is not sufficient to ensure that a complex plane cubic system has a complex center. This is done by finding a complex cubic system with a high order weak focus using an extensive computer search.

A Clebsch–Gordan formula for SL3(C) and applications to rationality☆

Abstract If R , S , T are irreducible SL 3 ( C ) -representations, we give an easy and explicit description of a basis of the space of equivariant maps R ⊗ S → T (Theorem 3.1). We apply this method to the rationality problem for invariant function fields. In particular, we prove the rationality of the moduli space of plane curves of degree 34. This uses a criterion which ensures the stable rationality of some quotients of Grassmannians by an SL-action (Proposition 5.4).

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

Let

Rationality of the moduli spaces of plane curves of sufficiently large degree

\cM_{7,n}

Some properties of dynamical degrees with a view towards cubic fourfolds

be the (coarse) moduli space of smooth curves of genus

On the rationality of the moduli space of Lüroth quartics

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