Dynamical systems associated with QRT families of degree four biquadratic curves each of them with genus zero

Abstract

We give some examples of QRT families of biquadratic curves which are of degree 4 (with high degree term $x^2y^2$), but such that every curve of such a family is singular, with genus 0, or is reducible (for some finite set of values of the parameter). This contrasts with classical examples of QRT families studied by many authors. We give also examples of QRT families of degree 4 whose every curve is reducible. Then we sketch out to study the dynamical systems associated to such QRT families of genus zero.