  We sharpen the two main tools used to treat the compactified Jacobian of a
singular curve: Abel maps and presentation schemes. First we prove a smoothness
theorem for bigraded Abel maps. Second we study the two complementary
filtrations provided by the images of certain Abel maps and certain
presentation schemes. Third we study a lifting of the Abel map of bidegree
(m,1) to the corresponding presentation scheme. Fourth we prove that, if a
curve is blown up at a double point, then the corresponding presentation scheme
is a IP^1-bundle. Finally, using Abel maps of bidegree (m,1), we characterize
the curves having double points at worst
