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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.

##### MSC:
 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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