  We analyze the unbinding transition for a two dimensional lattice polymer in
which the constituent strands are mutually avoiding random walks. At low
temperatures the strands are bound and form a single self-avoiding walk. We
show that unbinding in this model is a strong first order transition. The
entropic exponents associated to denaturated loops and end-segments
distributions show sharp differences at the transition point and in the high
temperature phase. Their values can be deduced from some exact arguments
relying on a conformal mapping of copolymer networks into a fluctuating
geometry, i.e. in the presence of quantum gravity. An excellent agreement
between analytical and numerical estimates is observed for all cases analized.
