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Covers of the projective line and the moduli space of quadratic differentials. (English) Zbl 1271.14036
The paper considers the \(1\)-dimensional Hurwitz spaces \(\overline{\mathcal{H}}_d({\mathbf c})\) of coverings \(\pi: C \to \mathbb{P}^1\) with prescribed branching behaviour at \(4\) points and no branching otherwise. They are finite coverings of \(\overline{M}_{0,4}\) and admit natural maps \(h:\overline{\mathcal{H}}_d({\mathbf c}) \to \overline{\mathcal{M}}_g\) that associate to \(\pi\) the source curve \(C\). The author proves an expression for the slope of the image under \(h\) of the irreducible components \(\overline{Z}_\mathcal{O}\) of \(\overline{\mathcal{H}}_d({\mathbf c})\), involving only combinatorial data. Examples are given, and the case of cyclic covers, where the monodromy datum is of the form \((\gamma^{a_1}, \gamma^{a_2}, \gamma^{a_3}, \gamma^{a_4})\) with \(\gamma \in S_d\) a cycle of length \(d\), is then worked out explicitly. The curves \(\overline{Z}_\mathcal{O}\) are Teichmüller curves, and the author derives a formula for the sum of their positive Lyapunov exponents (which was obtained previously in [G. Forni, C. Matheus and A. Zorich, J. Mod. Dyn. 5, No. 2, 285–318 (2011; Zbl 1276.37021)]).
Attention is then given to a special Hurwitz space \(\overline{\mathcal{H}}_d({\mathbf c})\), where the ramification profile \({\mathbf c}\) is chosen in such a way that the source curve admits a quadratic differential having only zeroes of odd order. These covers exist for any even degree, and an expression for its asymptotic slope as \(d \to \infty\) is given which depends on the Siegel-Veech constant of the corresponding stratum in the moduli space of quadratic differentials. Assuming that (as suggested by numerical evidence) these constants tend to \(1\) if the genus \(g\) tends to infinity, this gives an asymptotic lower bound of \(432/7g\) for the slope of the effective cone of \(\overline{\mathcal{M}}_g\) (cf. [D. Chen, G. Farkas and I. Morrison, “Effective divisors on moduli spaces of curves and abelian varieties”, arXiv:1205.6138]). As an application, this would give new asymptotic relations on the locus of Jacobians in the moduli space of abelian varieties.

MSC:
14H10 Families, moduli of curves (algebraic)
14H30 Coverings of curves, fundamental group
30F30 Differentials on Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37F99 Dynamical systems over complex numbers
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