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.
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.
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Ben Howard 1; John Millson 2; Andrew Snowden 3; Ravi Vakil 4
@article{CRMATH_2009__347_19-20_1177_0, author = {Ben Howard and John Millson and Andrew Snowden and Ravi Vakil}, title = {The relations among invariants of points on the projective line}, journal = {Comptes Rendus. Math\'ematique}, pages = {1177--1182}, publisher = {Elsevier}, volume = {347}, number = {19-20}, year = {2009}, doi = {10.1016/j.crma.2009.07.013}, language = {en}, }
TY - JOUR AU - Ben Howard AU - John Millson AU - Andrew Snowden AU - Ravi Vakil TI - The relations among invariants of points on the projective line JO - Comptes Rendus. Mathématique PY - 2009 SP - 1177 EP - 1182 VL - 347 IS - 19-20 PB - Elsevier DO - 10.1016/j.crma.2009.07.013 LA - en ID - CRMATH_2009__347_19-20_1177_0 ER -
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/
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