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Hyperelliptic classes are rigid and extremal in genus two. (English) Zbl 1442.14105
Summary: 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 D. Chen and N. Tarasca [Algebra Number Theory 10, No. 9, 1935–1948 (2016; Zbl 1354.14048)] and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension.
MSC:
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H30 Coverings of curves, fundamental group
14H10 Families, moduli of curves (algebraic)
14H45 Special algebraic curves and curves of low genus
14C25 Algebraic cycles
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