Segre surfaces and geometry of the Painlev\'e equations

TL;DR

This paper explores a family of affine Segre surfaces linked to the sixth Painlevé equation, revealing how various limits produce surfaces isomorphic to monodromy manifolds of different Painlevé equations.

Contribution

It establishes a geometric framework connecting Segre surfaces with Painlevé equations and their monodromy manifolds, providing new insights into their structure.

Findings

Affine Segre surfaces relate to Painlevé equations

Limiting forms produce isomorphic monodromy manifolds

Geometric perspective on Painlevé equations

Abstract

In this paper, we consider a six parameter family of affine Segre surfaces embedded in $\mathbb C^6$. For generic values of the parameters, this family is associated to the $q$-difference sixth Painlev\'e equation. We show that different limiting forms of this family give Segre surfaces that are isomorphic as affine varieties to the the monodromy manifolds of each Painlev\'e differential equation.