  We study the phase diagram of a one-dimensional Hubbard model where, in
addition to the standard nearest neighbor hopping $t$, we also include a
next-to-nearest neighbor hopping $t'$. For strong enough on-site repulsion,
this model has a transition at half filling from a magnetic insulator with
gapless spin excitations at small $t'/t$ to a dimerized insulator with a spin
gap at larger $t'/t$. We show that upon doping this model exhibits quite
interesting features, which include the presence of a metallic phase with a
spin gap and dominant superconducting fluctuations, in spite of the repulsive
interaction. More interestingly, we find that this superconducting phase can be
reached upon hole doping the magnetic insulator. The connections between this
model and the two chain models, recently object of intensive investigations,
are also discussed.
