The Teichmüller space of a punctured surface represented as a real algebraic space (Q1907013)

Suppose given an oriented closed surface of genus \(g\) with a set of \(s\) distinguished points \(P = \{x_1, \dots, x_s\}\). The surface may be triangulated by curves with endpoints in \(P\). If one considers a complete hyperbolic metric on the surface with singularities at the \(x_i\) giving angles of \(2 \pi/ \nu_i\), with the \(\nu_i\) natural numbers and each \(\nu_i \geq 2\), one can use the lengths of the curves of the triangulation (or their geodesic representatives) as coordinates for this metric as a point of Teichmüller space. These coordinates, suitably modified to give so-called \(L\)-length coordinates, give, as the authors prove, a representation of Teichmüller space as a real algebraic subspace of Euclidean space. By taking limits, the authors can also deal with the case where the points are not branch points but rather punctures. The authors make explicit calculations of the coordinates in the case of the twice-punctured torus.