Lutz Hille

Handbook of Tilting Theory: Fourier-Mukai transforms

In this paper we discuss some of the recent developments on derived equivalences in algebraic geometry.

Exceptional sequences of invertible sheaves on rational surfaces

In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.

A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical

LetG be a classical algebraic group defined over an algebraically closed field. We classify all instances when a parabolic subgroupP ofG acts on its unipotent radicalPu, or onpu, the Lie algebra ofPu, with only a finite number of orbits.The proof proceeds in two parts. First we obtain a reduction to the case of general linear groups. In a second step, a solution for these is achieved by studying the representation theory of a particular quiver with certain relations.Furthermore, for general linear groups we obtain a combinatorial formula for the number of orbits in the finite cases.

A counterexample to King's conjecture

Kings conjecture states that on every smooth complete toric variety

Strong Exceptional Sequences Provided by Quivers

X

The Δ-Filtered Modules Without Self-Extensions for the Auslander Algebra of k[T]/〈T n 〉

there exists a strongly exceptional collection which generates the bounded derived category of

Hochschild (co-)homology of schemes with tilting object

X

On the volume of a tilting module

and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface

On parabolic subgroups of classical groups with a finite number of orbits on the unipotent radial

\mathbb{F}_2

Quivers, cones and polytopes

iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7.