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Pentagonal structure of the configuration space of five points in the real projective line. (English) Zbl 0919.52013
The configuration space of five points in the real projective line \(P^1\) is defined as the orbit space \(X^\circ\) of the diagonal action of PGL\(_2{\mathbb R}\) on the set \(\{ (x_1,\ldots,x_5)\in(P^1)^5\mid x_i \not= x_j\) for \(i\not=j \}\). The orbits of all \(5\)-tuples \((x_1,\ldots,x_5)\) such that no three of the \(x_i\) coincide form a smooth compactification \(X\) of \(X^\circ\). This space is the union of \(12\) pentagons which are glued along their edges and their vertices. These pentagons form a \(\{5,4\}\)-tesselation of \(X\).
The two-fold unbranched covering \(\widetilde{X}\) of \(X\) equals Poncet’s great dodecahedron which is a orientable surface of genus \(4\). Identifying antipodal points of \(\widetilde{X}\) one recovers \(X\) as a non-orientable closed surface of genus \(5\). Moreover, lifting the tesselation of \(X\) to \(\widetilde{X}\) yields a \(\{5,4\}\)-tesselation of the great dodecahedron by \(24\) pentagons.
Finally, the authors give a modular interpretation of \(X\) as a quotient of the real \(2\)-ball.

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
51M15 Geometric constructions in real or complex geometry
14P99 Real algebraic and real-analytic geometry
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