Contractible Extremal Rays on \overline{M}_{0,n}

TL;DR

This paper characterizes the extremal rays of the cones of curves and divisors on moduli spaces of stable rational curves and their quotients, providing explicit generators and structural descriptions for small n.

Contribution

It identifies generators for contractible extremal rays of the cones of curves and divisors on these moduli spaces, revealing the simplicial nature of the divisor cone on the quotient space.

Findings

Complete descriptions of NE_1(M_n) for n<8

Complete descriptions of NE_1(Q_n) for n<11

The cone of divisors NE^1(Q_n) is simplicial

Abstract

We consider the cones of curves and divisors on the moduli space of stable pointed rational curves,M_n, and on the quotient by the symmetric group, Q_n, which is a moduli space of pairs. We find generators for contractible extremal rays of the cone of curves NE_1(M_n), and for the cone of divisors NE^1(Q_n). This second cone turns out to be simplicial. We give complete descriptions of NE_1(M_n) and NE_1(Q_n) for small n (< 8 in the first case, < 11 in the second). We also have results of independent interest on when curves in a divisor generate the cone of curves of the ambient variety.