  The Poland-Scheraga model describes the denaturation transition of two
complementary - in particular, equally long - strands of DNA, and it has
enjoyed a remarkable success both for quantitative modeling purposes and at a
more theoretical level. The solvable character of the homogeneous version of
the model is one of features to which its success is due. In the bio-physical
literature a generalization of the model, allowing different length and non
complementarity of the strands, has been considered and the solvable character
extends to this substantial generalization. We present a mathematical analysis
of the homogeneous generalized Poland-Scheraga model. Our approach is based on
the fact that such a model is a homogeneous pinning model based on a bivariate
renewal process, much like the basic Poland-Scheraga model is a pinning model
based on a univariate, i.e. standard, renewal. We present a complete analysis
of the free energy singularities, which include the localization-delocalization
critical point and (in general) other critical points that have been only
partially captured in the physical literature. We obtain also precise estimates
on the path properties of the model.
