Discrete Painlev\'e equations from pencils of quadrics in $\mathbb P^3$ with branching generators

In this paper we extend the novel approach to discrete Painlev\'e equations initiated in our previous work [2]. A classification scheme for discrete Painlev\'e equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in $\mathbb P^3$. A discrete Painlev\'e equation is viewed as an autonomous transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial $\Delta(\lambda)$ of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlev\'e map corresponds to a translation on the universal cover of the Riemann surface of $\sqrt{\Delta(\lambda)}$, rather than to a M\"obius transformation of the pencil parameter $\lambda$ as in [2].