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.

Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver Q it is thus desirable to decide if the cluster algebra defined by Q is acyclic. We call Q cluster-acyclic in this case, otherwise cluster-cyclic. In this note we classify the clustercyclic quivers with three vertices using a Diophantine equation studied by Markov.

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King's conjecture states that on every smooth complete toric variety X there exists a strongly exceptional collection which generates the bounded derived category of X and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface F 2 iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7 which consist of line bundles.

Let P be a parabolic subgroup of some general linear group GL V where V is a finite-dimensional vector space over an infinite field. The group P acts by conjugation on its unipotent radical P u and via the adjoint action on u , the Lie algebra of P u . More generally, we consider the action of P on the lth member of the descending central series of u , denoted by l u . Let u denote the nilpotency class of P u . In our main result we show that P acts on l u with a finite number of orbits precisely when u ≤ 4 for l = 0, or u ≤ 5 + 2l for l ≥ 1. Moreover, in case the field is algebraically closed, we consider the modality mod P l u of the action of P on l u . We show that mod P l u grows linearly in the minimal cases which admit infinitely many orbits (i.e., u = 5 for l = 0, or u = 6 + 2l for l ≥ 1), whereas the corresponding modality grows quadratically in all other infinite cases. These results are obtained by interpreting the orbits of P on l u as isomorphism classes of good modules over certain quasi-hereditary algebras and by a detailed inspection of the -representation types of these algebras.