The Cox ring of $\overline {M}_{0,6}$

Abstract:

We prove that the Cox ring of the moduli space $\overline {M}_{0,6}$, of stable rational curves with $6$ marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are pull-backs of the hyperelliptic locus in $\overline {M}_3$ via morphisms $\rho :\overline {M}_{0,6}\rightarrow \overline {M}_3$ that send a $6$-pointed rational curve to a curve with $3$ nodes by identifying $3$ pairs of points. In particular this gives a self-contained proof of Hassett and Tschinkel’s result about the effective cone of $\overline {M}_{0,6}$ being generated by the above mentioned divisors.