A study of the irreducibility of a generic hyperplane section through a normal singularity

E. García Barroso, P. González Pérez and P. Popescu-Pampu stated as an open problem the characterization of normal surface singularities whose generic hyperplane sections are irreducible. We show that this condition may be reformulated in terms of the maximal cycle of a resolution and we exhibit a class of singularities which satisfy it and whose dual graphs are trees. We generalize some results to higher dimensions and study when generic hyperplane sections through some normal $n$-dimensional singularities are irreducible. We also generalize a theorem of Michael Artin from rational singularities to normal singularities satisfying certain conditions.