Curves with only triple ramification.
Let \(C\) be a smooth proper curve of genus \(g\) over an algebraically closed field \(k\) of characteristic \(p \geq 0\). In the paper under review, the author poses the following question: Does there exist a generically étale finite map \(C \to {\mathbb P}^{1}\) whose ramification points all have index \(e=3\)? The main result of this paper is the following: Suppose \(p \neq 3\) and assume \(g \geq 2\). Let \(S\subset M_{g}\) be the closure of the set of all closed points such that the corresponding curve \(C\) admits a generically étale map \(C \to {\mathbb P}^{1}_{k}\) with only triple ramification. Then we have \(\dim(S) \geq 2g-3\). Furthermore, the author extends a result on elliptic complex curves of \textit{M. Fried, E. Klassen} and \textit{Y. Kopeliovich} [Proc. Am. Math. Soc., 129, 111--119 (2001; Zbl 0978.30026)] to all characteristics \(p \neq 3\), proving that for all but finitely many \(j\)-invariants \(j \in k\), the corresponding elliptic curve \(E\) admits a finite generically étale map \(C \to {\mathbb P}^{1}_{k}\) with only triple ramification. For the proof, the author uses deformation theory and the moduli space of stable curves.
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