Extremality of Rational Tails Boundary Strata in $\overline{\mathcal{M}}_{g,n}$

TL;DR

This paper investigates the extremality of boundary strata of rational tails in the moduli space of curves, providing evidence that all such strata are extremal in their effective cones, especially in genus zero.

Contribution

It develops techniques to analyze effective cones and proves extremality of boundary strata of rational tails, supporting the conjecture that all boundary strata are extremal.

Findings

Many boundary strata of rational tails are extremal in their effective cones.

All boundary strata are extremal in genus zero.

Provides techniques for studying effective cones of higher codimension classes.

Abstract

We review and develop some techniques used to investigate the effective cones of higher codimension classes. Our results show that a large collection of boundary strata of rational tails type are extremal in their effective cones on $\overline{\mathcal{M}}_{g,n}$ and provide evidence for the conjecture that all boundary strata of $\overline{\mathcal{M}}_{g,n}$ are extremal. As a corollary, we show that all boundary strata are extremal in genus zero.