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Comptes Rendus Mathématique
Volume 347, n° 19-20
pages 1177-1182 (octobre 2009)
Doi : 10.1016/j.crma.2009.07.013
Received : 9 July 2009 ;  accepted : 17 July 2009
The relations among invariants of points on the projective line
Les relations entre invariants des points sur la droite projective
 

Ben Howard a , John Millson b , Andrew Snowden c , Ravi Vakil d
a Dept. of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA 
b Dept. of Mathematics, University of Maryland, College Park, MD 20742, USA 
c Dept. of Mathematics, Princeton University, Princeton, NJ 08544, USA 
d Dept. of Mathematics, Stanford University, Stanford, CA 94305, USA 

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).

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Résumé

Nous considérons l’anneau des invariants de n points ordonnés sur la droite projective. L’espace   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   et que les poids sont 16. Pour n inférieur ou égal à 6, la géométrie est classique. Le cas   est plus riche encore et est développé dans cet article. Pour citer cet article : B. Howard et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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