Duke Mathematical Journal

The equations for the moduli space of n points on the line

Benjamin Howard, John Millson, Andrew Snowden, and Ravi Vakil

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A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory (GIT) 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 article deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any weighting of the points, 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. We also show that the ideal of relations is generated in degree at most four, and we give an explicit description of the generators. If all the weights are even (e.g., in the case of equal weight for odd n), then we show that the ideal of relations is generated by quadrics

Article information

Duke Math. J., Volume 146, Number 2 (2009), 175-226.

First available in Project Euclid: 5 January 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14L24: Geometric invariant theory [See also 13A50]
Secondary: 14D22: Fine and coarse moduli spaces 14H10: Families, moduli (algebraic)


Howard, Benjamin; Millson, John; Snowden, Andrew; Vakil, Ravi. The equations for the moduli space of $n$ points on the line. Duke Math. J. 146 (2009), no. 2, 175--226. doi:10.1215/00127094-2008-063. https://projecteuclid.org/euclid.dmj/1231170938

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