Extremal higher codimension cycles on moduli spaces of curves

TL;DR

This paper identifies and proves the extremality of various higher codimension cycles, including boundary strata and special loci, in the effective cone of moduli spaces of stable curves, revealing their geometric significance.

Contribution

It establishes the extremality of certain boundary strata and special loci in the effective cone of moduli spaces of curves, including infinitely many codimension two cycles.

Findings

Codimension two boundary strata are extremal.

Hyperelliptic loci with marked points are extremal.

Infinitely many extremal codimension two cycles exist in certain moduli spaces.

Abstract

We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space $\overline{\mathcal M}_{g,n}$ of stable genus $g$ curves with $n$ ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in $\overline{\mathcal M}_{3,1}$ and the locus of hyperelliptic curves in $\overline{\mathcal M}_4$ are extremal cycles. In addition, we exhibit infinitely many extremal codimension two cycles in $\overline{\mathcal M}_{1,n}$ for $n\geq 5$ and in $\overline{\mathcal M}_{2,n}$ for $n\geq 2$.