# Birational geometry of moduli spaces of configurations of points on the line

## TL;DR

This paper investigates the birational geometry of moduli spaces of n ordered points on the projective line, revealing the structure of their Mori cones and computing key invariants like the canonical divisor.

## Contribution

It characterizes the extremal rays of the Mori cone of these moduli spaces and develops tools to compute their canonical divisor, Hilbert polynomial, and automorphism group.

## Key findings

- Extremal rays of the Mori cone are generated by boundary strata.
- Computed the canonical divisor and Hilbert polynomial of the moduli space.
- Determined the automorphism group of the space.

## Abstract

In this paper we study the geometry of GIT configurations of $n$ ordered points on $\mathbb{P}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $(\mathbb{P}^1)^n//PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of automorphisms.