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The ring of evenly weighted points on the line

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Let \(M_w = ({\mathbb {P}}^1)^n /\!/\hbox {SL}_2\) denote the geometric invariant theory quotient of \(({\mathbb {P}}^1)^n\) by the diagonal action of \(\hbox {SL}_2\) using the line bundle \(\mathcal {O}(w_1,w_2,\ldots ,w_n)\) on \(({\mathbb {P}}^1)^n\). Let \(R_w\) be the coordinate ring of \(M_w\). We give a closed formula for the Hilbert function of \(R_w\), which allows us to compute the degree of \(M_w\). The graded parts of \(R_w\) are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights \(w_i\) are even, we find a presentation of \(R_w\) so that the ideal \(I_w\) of this presentation has a quadratic Gröbner basis. In particular, \(R_w\) is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of \(M_w\).

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Acknowledgments

We benefited from discussions with many people, including Federico Ardila, Aldo Conca, Sergey Fomin, Nathan Ilten, Chris Manon, Sam Payne, Bernd Sturmfels, and Ravi Vakil. We would also like to thank the referee, Diane Maclagan, and Burt Totaro for helpful comments on a previous version of this paper. Our main thanks is to Vic Reiner who was shaping the direction of this project. Part of this work was done at the Institute of Mathematics and its Applications, at the Mathematisches Forschungsinstitut Oberwolfach, and at the Max-Planck-Institut für Mathematik, and we would like to thank these institutes for providing a great research environment.

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Authors and Affiliations

  1. Maxwell Institute and School of Mathematics, University of Edinburgh, Edinburgh, UK

    Milena Hering

  2. Center for Communications Research, Institute for Defense Analysis, Princeton, NJ, 08540, USA

    Benjamin J. Howard

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  1. Milena Hering
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  2. Benjamin J. Howard
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Correspondence to Milena Hering.

Additional information

M. Hering was partially supported by an Oberwolfach Leibniz Fellowship and NSF Grant DMS 1001859.

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Hering, M., Howard, B.J. The ring of evenly weighted points on the line. Math. Z. 277, 691–708 (2014). https://doi.org/10.1007/s00209-013-1272-4

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