Triple coverings of algebraic surfaces according to the Cardano formula (Q1181048)

The general structure of a triple cover of an algebraic variety is described by \textit{R. Miranda} in Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011). A generalization of Miranda's result to the case of arbitrary schemes and the description of the characteristic 3 case are contained in the reviewer's paper in Ark. Mat. 27, No. 2, 319-341 (1989; Zbl 0707.14010). In the present paper the author tries a new approach to the study of triple covers. Given a triple cover \(p:X\to Y\), he considers its Galoisization \(\hat p:\hat X\to Y\), namely the smallest Galois cover of \(Y\) such that there exists \(\alpha:\hat X\to X\) which is a morphism over \(Y\), and tries to deduce properties of \(X\) from the properties of \(\hat X\). --- This way, he proves that a totally ramified triple cover of a simply connected variety is cyclic and describes \(\hat X\) and \(X\) in detail in the case that \(X\) is a surface and the singularities of the branch curve of \(p\) are cusps. In addition, several examples are carefully computed.