Special Year Seminar II

We present a combinatorial approach to the Deligne-Mumford-Knudsen compactification of the moduli space of n distinct points on the projective line $P^1$. 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 $X_n$ of the image there. Defining equations for $X_n$ are found via cross-ratios, and phylogenetic trees naturally yield a stratification. The forgetful map $X_{n+1} -> X_n$ exhibits as fibers n-pointed stable curves, which thus arise a posteriori and for free.

If time permits, the intriguing case of points in $P^2$ will be sketched.