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This paper deals with the geometry of the space [Geometric Invariant Theory (GIT) quotient] M8 of eight points in ℙ1, and the Gale-quotient N′8 of the GIT quotient of eight points in ℙ3.The space M8 comes with a natural embedding in ℙ13, or more precisely, the projectivization of the 8-representation V4, 4. There is a single 8-skew cubic  in ℙ13. The fact that M8 lies on the skew cubic  is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M8 is the singular locus of . 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 ℙ1 in general [20].The space N′8 comes with a natural embedding in ℙ13, or more precisely, ℙ V2, 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(M8) and the contraction Sec(M8) → N′8 factors through N8 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]). [ABSTRACT FROM PUBLISHER]

Consider the projective coordinate ring of the Geometric Invariant Theory (GIT) quotient (ℙ1) n //SL(2), with the usual linearization, where n is even. In 1894, Kempe proved that this ring is generated in degree one. In [2] we showed that, over ℚ, the relations between degree one invariants are generated by a class of quadratic relations—the simplest binomial relations—with the exception of n = 6, where there is a single cubic relation. The purpose of this article is to show that these results hold over , and to suggest why they may be true over . [ABSTRACT FROM PUBLISHER]

The ring of projective invariants of n ordered points on the projective line is one of the most basic and earliest studied examples in Geometric Invariant Theory. It is a remarkable fact and the point of this paper that, unlike its close relative the ring of invariants of n unordered points, this ring can be completely and simply described. In 1894 Kempe found generators for this ring, thereby proving the First Main Theorem for it (in the terminology introduced by Weyl). In this paper we compute the relations among Kempe's invariants, thereby proving the SecondMain Theorem (again in the terminology of Weyl), and completing the description of the ring 115 years later. This paper introduces a number of new tools to the problem, and uses the graphical algebra formalismto intermediate between representation-theoretic arguments (for symmetric and Lie groups), and the symmetry-breaking of the Speyer-Sturmfels degeneration. [ABSTRACT FROM AUTHOR]

Abstract: We consider the ring of invariants of n points on the projective line. The space is perhaps the first nontrivial example of a GIT quotient. The construction depends on the weighting of the n points. Kempe found generators (in the unit weight case) in 1894. We describe the full ideal of relations for all weightings. In some sense, there is only one equation, which is quadratic except for the classical case of the Segre cubic primal, for and weight 16. The cases of up to 6 points are long known to relate to beautiful familiar geometry. The case of 8 points turns out to be richer still. To cite this article: B. Howard et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: We use a simple description of the outer automorphism of to cleanly describe the invariant theory of six points in , , and . [Copyright &y& Elsevier]