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Extremal effective divisors on \(\overline{\mathcal M}_{1,n}\). (English) Zbl 1307.14034
The authors construct for every \(n \geq 3\) an infinite series of mutually non-proportional irreducible extremal effective divisors on \(\overline{\mathcal{M}}_{1,n}\), the Deligne-Mumford moduli space of pointed stable genus one curves. This shows that the pseudoeffective cone of this space is not a finitely generated polyhedral cone and hence that \(\overline{\mathcal{M}}_{g,n}\) is not a Mori Dream Space for \(g = 1\) and \(n \geq 3\). This was previously known for \(g \geq 3\) and \(n \geq 1\) [S. Keel, Ann. Math. (2) 149, No. 1, 253–286 (1999; Zbl 0954.14004)] as well as for \(g = 0\) and \(n \geq 134\) [A.-M. Castravet and J. Tevelev, “\(\overline{M}_{0,n}\) is not a Mori Dream Space”, Preprint, arXiv:1311.7673].
For a non-trivial tuple of integers \(\mathbf{a} = (a_1, \dots, a_n)\), the divisor \(D_\mathbf{a}\) considered by the authors is given as the closure of the locus of smooth pointed curves \((E; p_1, \dots, p_n) \in \mathcal{M}_{1,n}\) such that \(\sum_{i=1}^n a_i p_i = 0\) in the Jacobian of \(E\) (or equivalently, under the group law on \(E\) itself). The authors show that if \(d = \text{gcd}(a_1, \dots, a_n)\), the divisor \(D_\mathbf{a}\) has one irreducible component for every positive integer dividing \(d\). In particular, it is irreducible when \(d = 1\). By reducing to the case \(n = 3\) and exhibiting a suitable testing curve the authors show that it is also extremal when \(\mathbf{a} = (a_1, a_2, -a_1 - a_2, 0, \dots, 0)\). By computing the class of \(D_\mathbf{a}\) (which has previously appeared in the literature), the authors show that varying \(a_1\) and \(a_2\) yields infinitely many mutually non-proportional divisor classes.
The result is supplemented by a discussion of the spaces \(\overline{\mathcal{M}}_{1,n} / G\), where \(G\) is a subgroup of \(S_n\) acting on the marked points. The authors show that the pseudoeffective cone remains non-finite polyhedral as long as the action of \(G\) has at least three orbits, while it becomes simplicial (spanned by boundary divisors) if \(G\) is the full symmetric group. It is unknow what happens in the inbetween cases when \(G\) has fewer than three orbits but is not the whole symmetric group.
In an appendix it is shown that a canonical form defined on the smooth locus of the coarse moduli scheme \(\overline{M}_{1,n}\) extends holomorphically to any resolution of it, a result that enables one to compute its Kodaira dimension by working on the space itself instead of a desingularization. This computation has no direct bearing on the main content of the paper, but is included by the authors as it is not readily accessible in the literature.

MSC:
14H10 Families, moduli of curves (algebraic)
14E30 Minimal model program (Mori theory, extremal rays)
14H52 Elliptic curves
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[1] Arbarello, E; Cornalba, M, The Picard groups of the moduli spaces of curves, Topology, 26, 153-171, (1987) · Zbl 0625.14014
[2] Belorousski, P.: Chow rings of moduli spaces of pointed elliptic curves. Thesis, The University of Chicago (1998) · Zbl 1066.14030
[3] Bini, G; Fontanari, C, Moduli of curves and spin structures via algebraic geometry, Trans. Amer. Math. Soc., 358, 3207-3217, (2006) · Zbl 1105.14030
[4] Bini, G; Fontanari, C; Viviani, F, Moduli of curves and spin structures via algebraic geometry, Int. Math. Res. Not. IMRN, 4, 740-780, (2012) · Zbl 1246.14038
[5] Boissy, C: Connected components of the strata of the moduli space of meromorphic differentials, preprint, arXiv:1211.4951 · Zbl 1323.30060
[6] Castravet, A-M; Tevelev, J, Hypertrees, projections and moduli of stable rational curves, J. Reine Angew. Math., 675, 121-180, (2013) · Zbl 1276.14040
[7] Castravet, A.-M., Tevelev, J.: \(\bar{M}_{0, n}\) is not a more dream space, preprint, arXiv:1311.7673 · Zbl 1322.14047
[8] Cautis, S.: Extending families of curves: monodromy and applications. Thesis, Harvard University (2006) · Zbl 1215.14024
[9] Cavalieri, R; Marcus, S; Wise, J, Polynomial families of tautological classes on \({\cal M}_{g, n}^{rt}\), J. Pure Appl. Algebra, 216, 950-981, (2012) · Zbl 1273.14053
[10] Chiodo, A., Farkas, G.: Singularities of the moduli space of level curves, preprint, arXiv:1205.0201 · Zbl 1398.14032
[11] Eisenbud, D; Harris, J, The Kodaira dimension of the moduli space of curves of genus \(\ge 23\), Invent. Math., 90, 359-387, (1987) · Zbl 0631.14023
[12] Farkas, G, Birational aspects of the geometry of \(\overline{{\cal M}}_g\), Surv. Differ. Geom., 14, 57-110, (2009) · Zbl 1215.14024
[13] Farkas, G; Verra, A, The classification of universal Jacobians over the moduli space of curves, Comm. Math. Helv., 88, 587-611, (2013) · Zbl 1322.14047
[14] Farkas, G; Verra, A, The universal theta divisor over the moduli space of curves, J. Math. Pures Appl. (9), 100, 591-605, (2013) · Zbl 1327.14131
[15] Grushevsky, S., Zakharov, D.: The double ramification cycle and the theta divisor. Duke Math. J. (to appear). arXiv:1206.7001 · Zbl 1327.14132
[16] Hain, R, Normal functions and the geometry of moduli spaces of curves, Handb. Modul., I, 527-578, (2013) · Zbl 1322.14049
[17] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. The Clarendon Press, Oxford University Press, New York (1979) · Zbl 0423.10001
[18] Harris, J, On the Kodaira dimension of the moduli space of curves II. the even-genus case, Invent. Math., 75, 437-466, (1984) · Zbl 0542.14014
[19] Harris, J., Morrison, I.: Moduli of curves, graduate texts in Mathematics 187. Springer, New York (1998)
[20] Harris, J; Mumford, D, On the Kodaira dimension of the moduli space of curves, Invent. Math., 67, 23-88, (1982) · Zbl 0506.14016
[21] Hu, Y; Keel, S, Mori dream spaces and GIT, Michigan Math. J., 48, 331-348, (2000) · Zbl 1077.14554
[22] Keel, S, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. Math. (2), 149, 253-286, (1999) · Zbl 0954.14004
[23] Keel, S; McKernan, J, Contractible extremal rays on \(\overline{M}_{0, n}\), Handb. Modul., II, 115-130, (2013) · Zbl 1322.14050
[24] Logan, A, The Kodaira dimension of moduli spaces of curves with marked points, Am. J. Math., 125, 105-138, (2003) · Zbl 1066.14030
[25] Ludwig, K, On the geometry of the moduli space of spin curves, J. Algebraic Geom., 19, 133-171, (2010) · Zbl 1248.14033
[26] Müller, F.: The pullback of a theta divisor to \(\overline{\cal M}_{g, n}\). Math. Nachr. 286(11-12), 1255-1266 (2013) · Zbl 1246.14038
[27] Pagani,: Chen-Ruan cohomology of \({\cal M}_{1, n}\) and \(\overline{{\cal M}}_{1, n}\). Ann. Inst. Fourier (Grenoble) 63(4), 1469-1509 (2013) · Zbl 0631.14023
[28] Pagani, N.: Moduli of abelian covers of elliptic curves, preprint, arXiv:1303.2991 · Zbl 1077.14554
[29] Smyth, DI, Modular compactifications of the space of pointed elliptic curves II, Compos. Math., 147, 1843-1884, (2011) · Zbl 1260.14033
[30] 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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