Howard, Ben; Millson, John; Snowden, Andrew; Vakil, Ravi The relations among invariants of points on the projective line. (English. Abridged French version) Zbl 1174.14039 C. R., Math., Acad. Sci. Paris 347, No. 19-20, 1177-1182 (2009). Summary: We consider the ring of invariants of \(n\) points on the projective line. The space \((\mathbb P^1)^n//\)SL\(_2\) is perhaps the first nontrivial example of a GIT quotient. The construction depends on the weighting of the \(n\) points. A. B. Kempe [Lond. M. S. Proc. XXV. 343–359 (1894; JFM 25.0235.01)] 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 \(n=6\) and weight \(1^{6}\). 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. Cited in 5 Documents MSC: 14L24 Geometric invariant theory 14N05 Projective techniques in algebraic geometry Citations:JFM 25.0235.01 PDFBibTeX XMLCite \textit{B. Howard} et al., C. R., Math., Acad. Sci. Paris 347, No. 19--20, 1177--1182 (2009; Zbl 1174.14039) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Riordan numbers: a(n) = (n-1)*(2*a(n-1) + 3*a(n-2))/(n+1). References: [1] Dolgachev, I.; Ortland, D., Point sets in projective space and theta functions, Astérisque, 165 (1988), 210 pp. (1989) · Zbl 0685.14029 [2] Freitag, E.; Salvati Manni, R., The modular variety of hyperelliptic curves of genus three, 2007, preprint [3] Howard, B.; Millson, J.; Snowden, A.; Vakil, R., The equations for the moduli space of \(n\) points on the line, Duke Math. J., 146, 2, 175-226 (2009) · Zbl 1161.14033 [4] B. Howard, J. Millson, A. Snowden, R. Vakil, The ideal of relations for the ring of invariants of \(n\); B. Howard, J. Millson, A. Snowden, R. Vakil, The ideal of relations for the ring of invariants of \(n\) · Zbl 1255.14039 [5] Kempe, A., On regular difference terms, Proc. London Math. Soc., 25, 343-350 (1894) · JFM 25.0235.01 [6] Speyer, D.; Sturmfels, B., The tropical Grassmannian, Adv. Geom., 4, 3, 389-411 (2004) · Zbl 1065.14071 [7] Weyl, H., The Classical Groups: Their Invariants and Representations (1997), Princeton U.P.: Princeton U.P. Princeton, NJ · Zbl 1024.20501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.