Hyperelliptic classes are rigid and extremal in genus two

TL;DR

This paper proves that certain classes of hyperelliptic curves with marked points are rigid and extremal in the effective cone of the moduli space of genus two curves, extending previous results to an infinite family.

Contribution

It establishes an infinite family of rigid and extremal classes in the effective cone of $ar{ ext{M}}_{2, ext{ell+2m+n}}$, generalizing prior work by Chen and Tarasca.

Findings

Hyperelliptic classes are shown to be rigid and extremal.

The results apply to arbitrary codimension in genus two.

Extends the class of known extremal classes in moduli spaces.

Abstract

We show that the class of the locus of hyperelliptic curves with $\ell$ marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free marked points is rigid and extremal in the cone of effective codimension-($\ell + m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work of Chen and Tarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension.