  We study complete exceptional collections of coherent sheaves over Del Pezzo
surfaces, which consist of three blocks such that inside each block all Ext
groups between the sheaves are zero. We show that the ranks of all sheaves in
such a block are the same and the three ranks corresponding to a complete
3-block exceptional collection satisfy a Markov-type Diophantine equation that
is quadratic in each variable. For each Del Pezzo surface, there is a finite
number of these equations; the complete list is given. 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
with the sum of ranks that is minimal for the solutions of the corresponding
Markov-type equation, and the orbits can be obtained from each other via
tensoring by an invertible sheaf and with the action of the Weyl group. This
allows us to compute the number of orbits up to twisting.
