  A conjecture of Bondal-Polishchuk states that, in particular for the bounded
derived category of coherent sheaves on a smooth projective variety, the action
of the braid group on full exceptional collections is transitive up to shifts.
We show that the braid group acts transitively on the set of maximal
numerically exceptional collections on rational surfaces up to isometries of
the Picard lattice and twists with line bundles. Considering the blow-up of the
projective plane in up to 9 points in very general position, these results lift
to the derived category. More precisely, we prove that, under these
assumptions, a maximal numerically exceptional collection consisting of line
bundles is a full exceptional collection and any two of them are related by a
sequence of mutations and shifts. The former extends a result of Elagin-Lunts
and the latter a result of Kuleshov-Orlov, both concerning del Pezzo surfaces.
In contrast, we show in concomitant work [Krah, A Phantom on a Rational
Surface, (Preprint)] that the blow-up of the projective plane in 10 points in
general position admits a non-full exceptional collection of maximal length
consisting of line bundles.
