Abstract
We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space M ¯ g , n of stable genus g curves with n ordered marked points. In particular, we prove that codimension 2 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 M ¯ 3 , 1 and the locus of hyperelliptic curves in M ¯ 4 are extremal cycles. In addition, we exhibit infinitely many extremal codimension 2 cycles in M ¯ 1 , n for n ⩾ 5 and in M ¯ 2 , n for n ⩾ 2 .
Highlights
We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space Mg,n of stable genus g curves with n ordered marked points
Let HP0 ⊂ HP be the closure of the locus of irreducible nodal hyperelliptic curves with a marked point
Let D12 ⊂ ∆1;{1} ∩ ∆1;∅ be the closure of the locus of a chain of three curves of genera 2, 0 and 1, respectively, such that the rational component contains the marked point
Summary
We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space Mg,n of stable genus g curves with n ordered marked points. (v) The locus of hyperelliptic curves with a marked Weierstrass point is a non-boundary extremal codimension two cycle in M3,1 (Theorem 4.6). (vi) The locus of hyperelliptic curves is a non-boundary extremal codimension two cycle in M4 (Theorem 5.7).
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