Birational Maps That Send Biquadratic Curves to Biquadratic Curves

Recently, many papers have begun to consider so-called non-QRT birational maps of the plane. Compared to the QRT family of maps which preserve each biquadratic curve in a fibration of the plane, non-QRT maps send a biquadratic curve to another biquadratic curve belonging to the same fibration or to a biquadratic curve from a different fibration of the plane. In this communication, we give the general form of a birational map derived from a difference equation that sends a biquadratic curve to another. The necessary and sufficient condition for such a map to exist is that the discriminants of the two biquadratic curves are the same (and hence so are the j-invariants). The result allows existing examples in the literature to be better understood and allows some statements to be made concerning their generality.