Characterizing singular curves in parametrized families of biquadratics

We consider families of biquadratic curves B = 0 on , defined with respect to arbitrarily many complex parameters. Due to the fact that these families include curve intersections across different parameter combinations, they represent a generalization of the non-intersecting foliations of one-parameter invariant curves associated with the QRT mapping. We use algebraic methods involving discriminants to provide a complete classification of the singular curves in these families. In developing this classification, we exploit the special symmetric nature of B; namely, that it is a quadratic in x and y whose reflection in the line y = x is given by a simple change of parameters. We also define a range of conditions in the biquadratic's parameters and demonstrate the manner in which they correspond to different geometric realizations of the singular curves.