Gonality for stable curves and their maps with a smooth curve as their target

Here we study the deformation theory of some maps f: X → ℙ ^r , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of with fixed topological type behaves like the stratification by gonality of M _g.