  In this note we apply the techniques of the toric systems introduced by
Hille-Perling to several problems on smooth projective surfaces: We showed that
the existence of full exceptional collection of line bundles implies the
rationality for small Picard rank surfaces; we proved equivalences of several
notions of cyclic strong exceptional collection of line bundles; we also
proposed a partial solution to a conjecture on exceptional sheaves on weak del
Pezzo surfaces.
