Extremal higher codimension cycles on moduli spaces of curves

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 $n\ge 5$ and in ${\overline{\mathcal{M}}}_{2,n}$ for $n\ge 2$.

25 pages

References in corpus (1)

• Mori's program for ${\bar{M}}_{0,6}$ with symmetric divisors

Cited by in corpus (13)

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• Morphisms and faces of pseudo-effective cones
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• Extremality of loci of hyperelliptic curves with marked Weierstrass points
• Cones of special cycles of codimension 2 on orthogonal Shimura varieties
• Loci of curves with subcanonical points in low genus
• Pseudo-effective cones of projective bundles and weak Zariski decomposition
• Extremality of Rational Tails Boundary Strata in ${\overline{\mathcal{M}}}_{g,n}$
• Extremal higher codimension cycles of the space of complete conics
• On the cone of effective 2-cycles on ${\overline{M}}_{0,7}$
• An extremal effective survey about extremal effective cycles in moduli spaces of curves
• Higher Codimension Cycles on the Hilbert Scheme of Three Points on the Projective Plane
• On the cone of effective surfaces on ${\overline{\mathcal{A}}}_3$