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This paper deals with the geometry of the space [Geometric Invariant Theory (GIT) quotient]  M  8  of eight points in ℙ 1 , and the Gale‐quotient  N ′ 8  of the GIT quotient of eight points in ℙ 3 .  The space  M  8  comes with a natural embedding in ℙ 13 , or more precisely, the projectivization of the  8 ‐representation  V  4, 4 . There is a single  8 ‐skew cubic  in ℙ 13 . The fact that  M  8  lies on the skew cubic  is a consequence of Thomae's formula for hyperelliptic curves, but more is true:  M  8  is the singular locus of . These constructions yield the free resolution of  M  8  and are used in the determination of the ‘single’ equation cutting out the GIT quotient of  n  points in ℙ 1  in general [20].  The space  N ′ 8  comes with a natural embedding in ℙ 13 , or more precisely, ℙ  V  2, 2, 2, 2 . There is a single skew quintic  containing  N ′ 8  and  N ′ 8  is the singular locus of the skew quintic .  The skew cubic  and skew quintic  are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec( M  8 ) and the contraction Sec( M  8 ) →  N ′ 8  factors through  N  8  via the space of eight points on a quadric surface. We conjecture (Conjecture 1.1) that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of  N ′ 8  (the closure of the union of quadrisecant  lines ), and that the quintic  is the trisecant variety. The resulting picture extends the classical duality in the 6‐point case between the Segre cubic threefold and the Igusa quartic threefold.  We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces (see for example [3, 5]).

The geometry of eight points in projective space: representation theory, Lie theory and dualities