Geometry of the space of monodromy data

TL;DR

This paper clarifies the foundational algebraic geometry aspects of the space of monodromy data related to q-Painlevé VI, building on prior work by Jimbo, Sakai, Joshi, and Roffelsen.

Contribution

It provides a rigorous algebraic geometric framework for the monodromy data space associated with q-Painlevé VI, resolving ambiguities in previous foundational treatments.

Findings

Established a sound algebraic geometric foundation for monodromy data space.

Clarified the connection between monodromy data and q-Painlevé VI.

Resolved ambiguities in previous foundational works.

Abstract

In a paper published by the Annales de la Facult\'e de Sciences de Toulouse, with Yousuke Ohyama, we defined and studied a space of monodromy data underlying the well known derivation of q-Painlev\'e VI equation from q-isomonodromy conditions by Jimbo and Sakai. In a recent ArXiv preprint, Nalini Joshi and Pieter Roffelsen pursued our work. However, both our article and their preprint are ambiguous on some foundational algebro-geometric matters. We proceed here to provide sound bases.