  We introduce the notion of exact tilting objects, which are partial tilting
objects $T$ inducing an equivalence between the abelian category generated by
$T$ and the category of modules over the endomorphism algebra of $T$.
  Given a chain of sufficiently negative rational curves on a rational surface,
we construct an exceptional sequence whose universal extension is an exact
tilting object. For a chain of (-2)-curves, we obtain an equivalence with
modules over a well known algebra.
