Discrete Painlevé equations and pencils of quadrics in ${\mathbb{P}}^3$

Abstract

Discrete Painlevé equations constitute a famous class of integrable non-autonomous second order difference equations. A classification scheme proposed by Sakai interprets a discrete Painlevé 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 Painlevé equations, where the family of generalized Halphen surfaces is replaced by a pencil of quadrics in ${\mathbb{P}}^3$. A discrete Painlevé 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 Möbius transformation of the pencil parameter. Thus, our scheme is based on the classification of pencils of quadrics in ${\mathbb{P}}^3$.

43 pp., 8 figures