Discrete Painlev\'e equations and pencils of quadrics in $\mathbb P^3$

Discrete Painlev\'e equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlev\'e equation as a birational map between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). We propose a novel geometric interpretation of discrete Painlev\'e equations, where 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 birational transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one, according to a M\"obius transformation of the pencil parameter. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$.