2006/07/16 by Benjamin Howard, Howard, Benjamin, John Millson +6
Mathematics · #14H10 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Primary 14L24 #Secondary 14D22 #math.AG #msc:14D22 #msc:14H10 #msc:14L24
paper · pdf · doi:10.48550/arxiv.math/0607372
25 pages, 16 figures
arxiv created 2006/07/16 · openalex publication_date 2006/07/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This note deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any linearization, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold (the space of six points with equal weight). Unlike many facts in geometric invariant theory, these results (at least for the stable locus) are field-independent, and indeed work over the integers.