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The real loci of the configuration space of six points on the projective line and a Picard modular 3-fold. (English) Zbl 0920.52003
The author studies the configuration space \(X(6)\) of six distinct points on the real projective line modulo projective equivalence. This space has 60 connected components and the author constructs three distinct compactifications of \(X(6)\), all of which are equivariant with respect to the action of the symmetric group \(S_6\). The first one is obtained by compactifying the 60 components to polyhedra and glueing them together along their faces. The other two arise from embeddings of \(X(6)\) into projective \(4\)-space and are equivalent to the Segre cubic and the Igusa quartic, respectively. The second compactification also admits a modular interpretation as a quotient of the real hyperbolic \(3\)-space.

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
14E05 Rational and birational maps