  We address the following question: Given a polarized toric surface (S,L), and
a general integral curve C of geometric genus g in the linear system |L|, do
there exist degenerations of C in |L| to general integral curves of smaller
geometric genera? We give an affirmative answer to this question for surfaces
associated to h-transverse polygons, provided that the characteristic of the
ground field is large enough. We give examples of surfaces in small
characteristic, for which the answer to the question is negative. In case the
answer is affirmative, we deduce that a general curve C as above is nodal. In
characteristic 0, we use the result to show irreducibility of Severi varieties
of a large class of polarized toric surfaces with h-transverse polygon.
