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

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.

22 pages (fixed one small inaccuracy)

References in corpus (5)

• On blowing up the weighted projective plane
• On the Fano variety of linear spaces contained in two odd-dimensional quadrics
• On the biregular geometry of the Fulton-MacPherson compactification
• On the birational geometry of spaces of complete forms II: skew-forms
• On the birational geometry of spaces of complete forms I: collineations and quadrics

Cited by in corpus (1)

• On the birational geometry of spaces of complete forms I: collineations and quadrics