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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.

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