Extremality of loci of hyperelliptic curves with marked Weierstrass points

Chen, Dawei; Tarasca, Nicola

doi:10.2140/ant.2016.10.1935

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Extremality of loci of hyperelliptic curves with marked Weierstrass points

Dawei Chen and Nicola Tarasca

Vol. 10 (2016), No. 9, 1935–1948

DOI: 10.2140/ant.2016.10.1935

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 $n \leq 6$ . 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 $n \leq 6$ . A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.

Keywords

subvarieties of moduli spaces of curves, effective cones, higher codimensional cycles

Mathematical Subject Classification 2010

Primary: 14H99

Secondary: 14C99

Milestones

Received: 9 November 2015

Revised: 17 June 2016

Accepted: 10 September 2016

Published: 22 November 2016

Authors

Dawei Chen
Department of Mathematics Boston College Maloney Hall 140 Commonwealth Ave Chestnut Hill, MA 02467-3806 United States

Nicola Tarasca
Department of Mathematics University of Utah 155 S 1400 E Salt Lake City, UT 84112 United States

Vol. 10, No. 9, 2016