Alex Massarenti

Moduli spaces of $n$ ordered points on the line are

constructed as GIT quotients of $(\mathbb{P}^1)^{n}$ by the diagonal

action of $PGL(2)$ with respect to any polarization. These spaces are

closely related to the Deligne-Mumford compactification

$\overline{M}_{0,n}$ of the moduli space of smooth rational curves

with $n$ ordered marked points.

A complete characterization of these GIT quotients in terms of linear

systems on $\mathbb{P}^{n-3}$ has been given by C. Kumar in terms of

suitable linear systems on $\mathbb{P}^{n-3}$. Thanks to Kumar

description we will manage to describe special arrangements of linear

spaces in these quotients, yielding interesting results on their

biregular geometry.

Furthermore, we will interpret the GIT quotient associated to the

symmetric polarization as a small transformation of the blow-up of

$\mathbb{P}^{n-3}$ at $n-1$ points, and we will determine its cones of

curves and divisors. Finally, we will see how classical and well-known

facts about the geometry of the Segre cubic, that is, the unique

(modulo automorphisms of $\mathbb{P}^4$) cubic hypersurface in $\mathbb{P}^4$ with ten nodes, descend from our results.

The talk will be given in french.