Segre surfaces as monodromy manifolds for all equations in Sakai’s classification

Determine whether Segre surfaces exist that serve as monodromy manifolds for every remaining equation in Sakai’s diagram (the classification of discrete Painlevé equations), extending the constructions proven here for q-Painlevé VI and the differential Painlevé equations.

Background

This paper constructs a six-parameter family of affine Segre surfaces associated with q-Painlevé VI and proves that in confluence limits these Segre surfaces are isomorphic (as affine varieties) to the known cubic monodromy manifolds of each differential Painlevé equation (except P^{D_8}). It further shows that the blow-down maps from the cubic monodromy manifolds to the Segre surfaces are Poisson, and that the Segre surfaces capture geometric structures such as lines linked to special solutions.

Sakai’s diagram classifies the full hierarchy of discrete Painlevé equations (elliptic, q-difference, and additive types). While this paper establishes Segre surfaces as monodromy manifolds for q-Painlevé VI and differential Painlevé equations, it leaves open whether analogous Segre-surface monodromy manifolds exist uniformly across all remaining equations in Sakai’s classification.