Extremality of loci of hyperelliptic curves with marked Weierstrass points

TL;DR

This paper proves that the classes of loci of genus-two curves with up to six marked Weierstrass points are extremal in the cone of effective classes, extending known results and constructing related extremal nef curve classes.

Contribution

It generalizes the extremality of loci of hyperelliptic curves with marked Weierstrass points for all n up to 6, providing new extremal classes in the effective cone.

Findings

Loci of genus-two curves with n marked Weierstrass points are extremal for n<=6.

The class of these loci spans an extremal ray in the effective cone.

Construction of extremal nef curve classes in moduli spaces of pointed elliptic curves.

Abstract

The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n<=6. It is well known that the class of the closure of the divisor obtained for n=1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the locus of genus-two curves with n marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension n, for n<=6. A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.