  The present paper is dedicated to the 2-dimensional Interacting Partially
Directed Self Avoiding Walk constrained to remain in the upper-half plan and
interacting with the horizontal axis. The model has been introduced in
\cite{F90} to investigate the behavior of a homopolymer dipped in a poor
solvent and adsorbed along a horizontal hard wall. It is known to undergo a
collapse transition between an extended phase, inside which typical
configurations of the polymer have a large horizontal extension (comparable to
their total size), and a collapsed phase inside which the polymer looks like a
globule.
  In the present paper, we establish rigorously that inside the collapsed
phase, a surface transition occurs between an adsorbed-collapsed regime where
the bottommost layer of the globule is pinned at the hard wall, and a
desorbed-collapsed regime where the globule wanders away from the wall. To
prove the existence of this surface transition and exhibit its associated
critical curve, we display some sharp asymptotics of the partition function for
a slightly simplified version of the model.
