The monodromy group of a function on a general curve (Q1885665)

By standard Brill-Noether theory, a general curve of genus \(g\) is a cover of \(\mathbb{P}^1\) of degree \(n\) if and only if \(g \leq 2(n-1)\). Moreover, if \(g \leq 2(n-1)\), one can find such a cover that is simple: this means that the monodromy of a simple loop around a branch point is a transposition, and in particular the monodromy group is the whole symmetric group \(S_n\). By a result of \textit{R. Guralnick} et al. [Contemp. Math. 186, 325--352 (1995; Zbl 0845.20002); J. Algebra 207, No.1, 127--145 (1998; Zbl 0911.20003); ``Symmetric and alternating groups as monodromy groups of Riemann surfaces. I'', preprint] for a general curve of genus \(g>3\), the monodromy group of a degree \(n\) cover onto \(\mathbb{P}^1\) is either the full permutation group \(S_n\) or the alternating subgroup \(A_n\) (assuming the cover does not factor non-trivially, i.e. assuming \(G\) to be a primitive subgroup of \(S_n\)). It was not known if the case \(A_n\) occurs. In the paper under review, an affirmative answer for large \(n\) is given. More precisely it is proved that, if \(g \geq 3\), the general curve of genus \(g\) admits a cover to \(\mathbb{P}^1\) of degree \(n\) with monodromy \(A_n\) such that all inertia groups are generated by double transpositions if and only if \(n \geq 2g+1\). It is moreover proved that the same result still holds after replacing double transpositions with 3-cycles. Both results (for 3-cycles and for double transpositions) are extended also to the case \(g=2\) (obtaining \(n\geq 6\)) and to the case \(g=1\) (obtaining \(n\geq 5\)).