On the effective cone of higher codimension cycles in $\overline{\mathcal{M}}_{g,n}$

TL;DR

This paper demonstrates the existence of infinitely many extremal effective cycles of higher codimension in the moduli space of stable curves, showing that the effective cone is not rational polyhedral in these cases.

Contribution

It constructs explicit examples of infinitely many extremal effective cycles in certain moduli spaces, revealing the non-polyhedral nature of their effective cones.

Findings

Infinite extremal effective cycles exist in specified cases

Effective cone is not rational polyhedral in these cases

Provides new insights into the geometry of ar{al M}_{g,n}

Abstract

We exhibit infinitely many extremal effective codimension-$k$ cycles in $\overline{\mathcal{M}}_{g,n}$ in the cases $g\geq 3, n\geq g-1$ and $k=2$, $g\geq 2$, $k\leq n-g,g,$ and $g=1$, $k\leq n-2$. Hence in these cases the effective cone is not rational polyhedral.