Minimality of the $\mathcal D$-groupoid of symmetries of a projective structure

TL;DR

This paper investigates the conditions under which Kummer's $\

Contribution

It establishes necessary and sufficient conditions for the minimality of Kummer's $\

Findings

Minimality relates to the strong minimality of the Schwarzian equation.

Conditions are linked to non-integrability by Liouvillian functions.

Results connect symmetry groupoids with differential equation properties.

Abstract

In this article we study Kummer's $\mathcal D$-groupoid, which is the groupoid of symmetries of a meromorphic projective structure. We give necessary and sufficient conditions for its minimality, in the sense of not having infinite sub-$\mathcal D$-groupoids. The condition that we find turns out to be equivalent to the strong minimality of the non-linear Schwarzian equation and the non-integrability by means of Liouvillian functions of the linear Schwarzian equation.