## 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)

Zbl 1078.14536

### Software:

M0nbar; Macaulay2
Full Text:

### References:

 [1] Carr, S.: A polygonal presentation of $${Pic(\overline{\mathfrak{M}}_{0,n})}$$ , (2009). arXiv:0911.2649 · Zbl 1358.14016 [2] Castravet, A.-M., Tevelev, J.: Rigid curves on $${\overline{M}_{0,n}}$$ and arithmetic breaks. Compact moduli spaces and vector bundles, pp. 19-67 (2012) · Zbl 1254.14028 [3] Castravet, A.-M.; Tevelev, J., Hypertrees, projections, and moduli of stable rational curves, J. Reine Angew. Math., 675, 121-180, (2013) · Zbl 1276.14040 [4] Castravet, A.-M., Tevelev, J.: $${\overline{M}_{0,n}}$$ is not a Mori Dream Space, 2013. arXiv:1311.7673, to appear in Duke Math. J. · Zbl 1343.14013 [5] Chen, D., Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228, 1135-1162, (2011) · Zbl 1227.14030 [6] Doran, B., Giansiracusa, N., Jensen, D.: A simplicial approach to effective divisors in $${\overline{M}_{0,n}}$$ , (2014) arXiv:1401.0350 · Zbl 1405.14020 [7] Gelfand I.M., Kapranov M.M., Zelevinsky A.V.: Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2008) · Zbl 1138.14001 [8] Grayson, D., Stillman, M.: Macaulay2 version 1.6. [9] Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math., 173, 316-352, (2003) · Zbl 1072.14014 [10] Kapranov, M.M.: Chow quotients of Grassmannians. I, I. M. Gelfand Seminar, pp. 29-110 (1993) · Zbl 0811.14043 [11] Keel, S., Intersection theory of moduli space of stable $$n$$-pointed curves of genus zero, Trans. Am. Math. Soc., 330, 545-574, (1992) · Zbl 0768.14002 [12] Keel, S., McKernan, J.: Contractible Extremal Rays on $${\overline{M}_{0,n}}$$ , (1996) arXiv:9607009 · Zbl 1072.14014 [13] Kollár, J.: Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer, Berlin (1996) · Zbl 0853.14020 [14] Kontsevich, M.; Manin, Yu., Gromov-Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys., 164, 525-562, (1994) · Zbl 0853.14020 [15] Losev, A.; Manin, Y., New moduli spaces of pointed curves and pencils of at connections, Mich. Math. J., 48, 443-472, (2000) · Zbl 1078.14536 [16] Moon, H.-B., Log canonical models for the moduli space of stable pointed rational curves, Proc. Am. Math. Soc., 141, 3771-3785, (2013) · Zbl 1358.14016 [17] Mumford D.: The Red Book of Varieties and Schemes, Expanded, Lecture Notes in Mathematics, vol. 1358. Springer, Berlin (1999) · Zbl 0945.14001 [18] Pandharipande, R., The canonical class of $${\overline{M}_{0,n}(P^r,d)}$$ and enumerative geometry, Int. Math. Res. Notices, 4, 173-186, (1997) · Zbl 0898.14010 [19] Vermeire, P., A counterexample to fulton’s conjecture on $${\overline M_{0,n}}$$, J. Algebra, 248, 780-784, (2002) · Zbl 1039.14014
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