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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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