The Ring of Evenly Weighted Points on the Line

Let Mw = (P)//SL2 denote the geometric invariant theory quotient of (P) by the diagonal action of SL2 using the line bundle O(w1, w2, . . . , wn) on (P). Let Rw be the coordinate ring of Mw. We give a closed formula for the Hilbert function of Rw, which allows us to compute the degree of Mw. The graded parts of Rw are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights wi are even, we find a presentation of Rw so that the ideal I of this presentation has a quadratic Gröbner basis. In particular, Rw is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of Mw.