The Geometry of Eight Points in Projective Space: Representation Theory, Lie Theory, Dualities

This paper deals with the geometry of the space (GIT quotient) M8 of 8 points in P, and the Gale-quotient N ′ 8 of the GIT quotient of 8 points in P. The space M8 comes with a natural embedding in P, or more precisely, the projectivization of the S8-representation V4,4. There is a single S8-skew cubic C in P. The fact that M8 lies on the skew cubic C is a consequence of Thomae’s formula for hyperelliptic curves, but more is true: M8 is the singular locus of C. These constructions yield the free resolution of M8, and are used in the determination of the “single” equation cutting out the GIT quotient of n points in P in general [HMSV4]. The space N ′ 8 comes with a natural embedding in P , or more precisely, PV2,2,2,2. There is a single skew quintic Q containing N ′ 8, and N ′ 8 is the singular locus of the skew quintic Q. The skew cubic C and skew quintic Q 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(M8), and the contraction Sec(M8) 99K N ′ 8 factors through N8 via the space of 8 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 Q 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, e.g. [CH], [B], etc.