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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.

14H10 Families, moduli of curves (algebraic)
14E30 Minimal model program (Mori theory, extremal rays)
14H52 Elliptic curves
Full Text: DOI
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