Affine coordinates for Teichmüller spaces (Q1105738)

\textit{H. Helling} has shown [Invent. Math. 17, 217-229 (1972; Zbl 0239.22015)] that the Teichmüller space of oriented compact Riemann surfaces without boundary can be represented as a component of an affine real algebraic variety. In this paper the construction of Helling is extended for compact surfaces which may have boundary components and which need not be orientable. It is shown that Teichmüller spaces of all compact surfaces (with negative Euler characteristic) can be represented as a component of an affine real algebraic variety. It is, furthermore, shown that, in the case of oriented surfaces without boundary, this affine real algebraic variety may be chosen in such a way that its codimension is 2. It is conjectured that this is the smallest possible codimension for such a representation of the Teichmüller space. These considerations have been motivated by the constructions of \textit{J. Morgan} and \textit{P. Shalen} [Ann. Math., II. Ser. 120, 401-476 (1984; Zbl 0583.57005)] who have shown that the real spectrum compactification of the affine variety corresponding to the Teichmüller space of compact and oriented surfaces gives the Thurston compactification of the Teichmüller space.