  Negative Sasakian manifolds, where the first Chern class of the contact
subbundle is a torsion class, can be viewed as Seifert-$S^1$ bundles where the
base orbifold has an ample orbifold canonical class. We use this framework to
settle completely an open problem formulated by C.Boyer and K.Galicki which
asks whether or not arbitrary connected sums of $S^2\times S^3$ are negative
Sasakian manifolds. As a consequence of the affirmative answer to this problem,
there exists so-called Sasaki eta-Einstein and Lorentzian Sasaki-Einstein
metrics on all of these five-manifolds and moreover all of these can be
realized as links of isolated hypersurface singularities defined by weighted
homogenous polynomials. The key step is to construct infinitely many
hypersurfaces with branch divisors in weighted projective that contain only
rational curves.
