  Let $M_w = (\Pj^1)^n \q \mathrm{SL}_2$ denote the geometric invariant theory
quotient of $(\Pj^1)^n$ by the diagonal action of $\mathrm{SL}_2$ using the
line bundle $\mathcal{O}(w_1,w_2,...,w_n)$ on $(\Pj^1)^n$. 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 $I$ of this presentation has a quadratic Gr\"obner
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 $M_w$.
