Comptes Rendus
Algebraic Geometry
The relations among invariants of points on the projective line
Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1177-1182.

We consider the ring of invariants of n points on the projective line. The space (P1)n//SL2 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 n=6 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.

Nous considérons l'anneau des invariants de n points ordonnés sur la droite projective. L'espace (P1)n//SL2 est peut-être le premier exemple intéressant d'un quotient GIT. La construction dépend du choix des poids pour les n points. En 1894, Kempe a introduit un ensemble de générateurs (dans le cas où tous les poids sont égaux à 1). Ici, nous décrivons les relations entre les générateurs pour tous les choix de poids. En un sens il n'y a qu'une relation, qui est quadratique sauf dans le cas classique de la cubique de Segre, lorsque n=6 et que les poids sont 16. Pour n inférieur ou égal à 6, la géométrie est classique. Le cas n=8 est plus riche encore et est développé dans cet article.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2009.07.013
Ben Howard 1; John Millson 2; Andrew Snowden 3; Ravi Vakil 4

1 Dept. of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
2 Dept. of Mathematics, University of Maryland, College Park, MD 20742, USA
3 Dept. of Mathematics, Princeton University, Princeton, NJ 08544, USA
4 Dept. of Mathematics, Stanford University, Stanford, CA 94305, USA
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Ben Howard; John Millson; Andrew Snowden; Ravi Vakil. The relations among invariants of points on the projective line. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1177-1182. doi : 10.1016/j.crma.2009.07.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.07.013/

[1] I. Dolgachev; D. Ortland Point sets in projective space and theta functions, Astérisque, Volume 165 (1988) 210 pp. (1989)

[2] E. Freitag; R. Salvati Manni The modular variety of hyperelliptic curves of genus three (2007, preprint) | arXiv

[3] B. Howard; J. Millson; A. Snowden; R. Vakil The equations for the moduli space of n points on the line, Duke Math. J., Volume 146 (2009) no. 2, pp. 175-226

[4] B. Howard, J. Millson, A. Snowden, R. Vakil, The ideal of relations for the ring of invariants of n points on the line, in preparation

[5] A. Kempe On regular difference terms, Proc. London Math. Soc., Volume 25 (1894), pp. 343-350

[6] D. Speyer; B. Sturmfels The tropical Grassmannian, Adv. Geom., Volume 4 (2004) no. 3, pp. 389-411

[7] H. Weyl The Classical Groups: Their Invariants and Representations, Princeton U.P., Princeton, NJ, 1997

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