Phylogenetic Trees and the Moduli of n Points

We present a combinatorial approach to the Deligne-Mumford-Knudsen compactification of the moduli space of n distinct points on the projective line P1. The idea is to choose a totally symmetric embedding of the orbits of generic points into a high-dimensional projective scheme and to take the Zariski-closure Xn of the image there. Defining equations for Xn are found via cross-ratios, and phylogenetic trees naturally yield a stratification. The forgetful map Xn+1−>Xn exhibits as fibers n-pointed stable curves, which thus arise a posteriori and for free.

If time permits, the intriguing case of points in P2

will be sketched.