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Extremal higher codimension cycles on moduli spaces of curves. (English) Zbl 1357.14039
Let \(\overline{\mathcal M}_{g, n}\) denote the Deligne-Mumford-Knudsen moduli space of stable genus \(g\) curves with \(n\) ordered marked points defined over the field of complex numbers. The authors prove the following:
(i) Every codimension 2 boundary stratum of \(\overline{\mathcal M}_{g}\) and of \(\overline{\mathcal M}_{0, n}\) is extremal.
(ii) The higher codimension boundary strata associated to certain dual graphs described in Sect. 5 are extremal in \(\overline{\mathcal M}_{g}\).
(iii) Every codimension \(k\) boundary stratum of \(\overline{\mathcal M}_{0, n}\) parameterizing curves with \(k\) marked tails attached to an unmarked \({\mathbb P}^1\) is extremal.
(iv) There exist infinitely many extremal effective codimension 2 cycles in \(\overline{\mathcal M}_{1, n}\) for every \(n\geqslant 5\) and in \(\overline{\mathcal M}_{2, n}\) for every \(n\geqslant 2\).
(v) The locus of hyperelliptic curves with a marked Weierstrass point in a nonboundary extremal codimension 2 cycle is \(\overline{\mathcal M}_{3, 1}\).
(vi) The locus of hyperelliptic curves is a nonboundary extremal codimension 2 cycle in \(\overline{\mathcal M}_{4}\).

MSC:
14H10 Families, moduli of curves (algebraic)
14C25 Algebraic cycles
14E30 Minimal model program (Mori theory, extremal rays)
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