The moduli space \( M \)\( _{0,n} \)-bar describes n-marked stable curves of genus zero. It has a closed embedding into a product of projective spaces \( P^1 \times\cdots\times P^{n-3} \), due to Keel-Tevelev and Kapranov and involving the  tautological "psi" and "omega" divisor classes. Composing this map with the Segre map gives the so-called "log canonical" embedding of \( M \)\( _{0,n} \)-bar. Essentially, these maps are the most natural ways to realize \( M \)\( _{0,n} \)-bar as a (multi)projective variety.

I will introduce \( M \)\( _{0,n} \)-bar and discuss some combinatorics and algebra arising from this embedding. Most notably, Monin and Rana (2017) conjectured a set of equations defining the embedded image of \( M \)\( _{0,n} \)-bar in \( P^1 \times \cdots \times P^{n-3} \), given by 2-by-2 minors of certain 2-by-k matrices, resembling the well-known equations for Veronese, Segre and Plücker embeddings. We prove this conjecture for all n.

This is joint work with Maria Gillespie and Sean T. Griffin.