Summary: This paper examines the Schwarz operator A and its relatives \(\dot A\), \(\overline A\) and that are assigned to a minimal surface \(X\) which maps consequtive arcs of the boundary of its parameter domain onto the straight lines which are determined by pairs \(P_{j}\), \(P_{j +1}\) of two adjacent vertices of some simple closed polygon \({\Gamma\subset \mathbb{R}^3}\). In this case \(X\) possesses singularities in those boundary points which are mapped onto the vertices of the polygon \(\Gamma\). Nevertheless it is shown that A and its closure \(\overline A\) have essentially the same properties as the Schwarz operator assigned to a minimal surface which spans a smooth boundary contour. This result is used by the author to prove in [R. Jakob, Finiteness of the set of solutions of Plateau’s problem for polygonal boundary curves. Bonn: Univ. Bonn, Mathematisches Institut (2006; Zbl 1109.53009)] the finiteness of the number of immersed stable minimal surfaces which span an extreme simple closed polygon \(\Gamma\), and in [Jakob (loc. cit.)] even the local boundedness of this number under sufficiently small perturbations of \(\Gamma\).