  In this article we introduce a new class of non-commutative projective curves
and show that in certain cases the derived category of coherent sheaves on them
has a tilting complex. In particular, we prove that the right bounded derived
category of coherent sheaves on a reduced rational projective curve with only
nodes and cusps as singularities, can be fully faithfully embedded into the
right bounded derived category of the finite dimensional representations of a
certain finite dimensional algebra of global dimension two. As an application
of our approach we show the dimension of the bounded derived category of
coherent sheaves on a rational projective curve with only nodal or cuspidal
singularities is at most two. In the case of the Kodaira cycles of projective
lines, the corresponding tilted algebras belong to a well-known class of gentle
algebras. We work out in details the tilting equivalence in the case of the
plane nodal cubic.
