On the Kodaira dimension of Hurwitz spaces (Q2118188)

The Hurwitz space \(\mathcal H_g^k\) is the parameter space of covers \([f: C \to \mathbb P^1, p_1, \dots, p_b]\) where \(C\) is a smooth genus \(g\) curve and \(f\) is a map of degree \(k\) simply branched over \(b = 2g+2k-2\) ordered distinct points \(p_1, \dots, p_b \in \mathbb P^1\). \textit{J. Harris} and \textit{D. Mumford} constructed the moduli space \(\overline{\mathcal H}^k_g\) of admissible covers [Invent. Math. 67, 23--88 (1982; Zbl 0506.14016)], which comes equipped with two natural maps \(\sigma: \overline{\mathcal H}^k_g \to \overline{\mathcal M}_g\) and \(\mathfrak b: \overline{\mathcal H}^k_g \to \overline{\mathcal M}_{0,b}\). The authors prove that the moduli \textit{stack} \(\overline H^k_g\) of degree \(k\) admissible covers has maximal Kodaira dimension for \(g \geq 2\) and \(k \geq 3\). The result is sharp because \(\overline{\mathcal H}^2_g\) is rational, hence the canonical class on both \(\overline{\mathcal H}^2_g\) and the stack \(\overline H^2_g\) is not effective. The proof comes from an analysis of the map \(\theta: \overline{\mathcal H}^k_g \to \overline{\mathcal M}_{g, b+b[k-2]}\) associating an admissible cover as above to the pointed curve \([C,x_1,\dots,x_b,A_1, \dots, A_b]\) where \(x_i \in f^{-1}(p_i)\) is the unique ramification point of \(f\) lying over the branch point \(p_i\) and \(A_i = f^{-1} (p_i) - \{x_i\}\) is the \(i\)th set of \textit{antiramification points} of \(f\). A similar argument proves that the coarse moduli space \(\overline{\mathcal H}^2_g\) has maximal Kodaira dimension for \(g \geq 2\). One does not expect a uniform result like the main theorem for Hurwitz spaces \(\overline{\mathcal H}_{g,k}\) where the branch points are unordered, see for example work of \textit{F. Geiß} [Doc. Math. 17, 627--640 (2012; Zbl 1266.14019)].