Positive divisors on quotients of $\bar{M}_{0,n}$ and the Mori cone of $\bar{M}_{g,n}$

TL;DR

This paper characterizes certain invariant divisors on moduli spaces of stable pointed curves, showing they are effective combinations of boundary divisors, and determines the Mori cone for small genus cases.

Contribution

It proves that for large enough symmetry, invariant F-nef divisors are effective boundary combinations, and computes the Mori cone for specific small-genus moduli spaces.

Findings

Invariant F-nef divisors are effective boundary combinations for m ≥ n-3.

The Mori cone of small genus moduli spaces is explicitly determined.

Provides new insights into the structure of divisors on moduli spaces.

Abstract

We prove that if $m \ge n-3$ then every $S_m$-invariant F-nef divisor on the moduli space of stable $n$-pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine the Mori cone of the moduli spaces of stable curves of small genus with few marked points.