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Effective curves on \(\overline{\mathrm{M}}_{0,n}\) from group actions. (English) Zbl 1329.14064

Let \(\overline{M}_{0,n}\) be the moduli space of stable \(n\)-pointed rational curves. There is a natural \(S_n\)-action on \(\overline{M}_{0,n}\) that permutes the marked points. For a subgroup \(G\) of \(S_n\), let \(\overline{M}_{0,n}^G\) be the union of irreducible components of the \(G\)-fixed locus that intersect the interior \(M_{0,n}\). Imposing suitable numerical conditions on \(G\), \(\overline{M}_{0,n}^G\) then becomes an irreducible curve on \(\overline{M}_{0,n}\). The aim of this paper is to compute the numerical class of \(\overline{M}_{0,n}^G\) and to determine whether it is an effective linear combination of one-dimensional boundary strata of \(\overline{M}_{0,n}\) (called F-curves). The authors come up with a method to approach this computational problem using Losev-Manin spaces [A. Losev and Y. Manin, Mich. Math. J. 48, 443–472 (2000; Zbl 1078.14536)] and tori degenerations. They demonstrate the method by focusing on the cases when \(G\) is either a cyclic group or a dihedral group. The outcome seems to suggest that the numerical classes of \(\overline{M}_{0,n}^G\) are always effective linear combinations of F-curves.

MSC:

14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 1078.14536

Software:

M0nbar; Macaulay2
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References:

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