Three-block exceptional collections over Del Pezzo surfaces

Abstract

We study complete exceptional collections of coherent sheaves over Del Pezzo surfaces that consist of three blocks such that all Ext groups between the sheaves inside each block are zero. We show that the ranks of all sheaves in such a block are equal, and that the three ranks corresponding to a complete 3-block exceptional collection satisfy a Markov-type Diophantine equation, which is quadratic in each variable. For each Del Pezzo surface, there are finitely many such equations, and we give a complete list of them. The 3-string braid group acts by mutations on the set of complete 3-block exceptional collections. We describe this action. In particular, any orbit contains a 3-block collection the sum of whose ranks is minimal for the solutions of the corresponding Markov-type equation, and the orbits can be obtained from each other under tensoring with an invertible sheaf and the action of the Weyl group. This enables us to compute the number of orbits up to twisting.