The ring of evenly weighted points on the line

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) .